Combinatorial Game Theory
who wins, and by how much
The computed corpus

The position index

Every position drawn anywhere on this site, with the value the solver computed for it and the essays that argue about it. Nothing here was typed in — the index is built by running each essay's own figure calls again and asking what came back.

A collection of essays about computed values ought to be able to show the values it has computed. This is that list: 1667 positions across 78 games, each with the value the recursion returned and the outcome class that follows from it.

It is not a list kept beside the essays. Every figure call in every essay is read out of the markdown and run again at exactly the options the essay passed, and what a generator says it drew is what goes in the table. An index maintained by hand drifts away from the figures as soon as one of them changes; this one cannot drift, because it is the same computation.

698 of these positions are drawn in more than one essay, which is the collection working as intended: a value computed once, then argued about from several directions. And 80 of the site's 146 generators declare positions at all — the others draw an argument rather than a position, a thermograph or a running-time curve or a reduction, and they are absent from this page rather than padded into it.

A form

2 positions

PositionWorthOutcomeDrawn in
{−1, 0, ∗ | 1} 1/2 L The reduction that always shrinks
{0, ∗ | ∗} ↑ L The reduction that always shrinks

Amazons

57 positions

PositionWorthOutcomeDrawn in
..LRxx.x. 0 | −2
territory count −1		 N When a real board falls apart
..Lxx..xR {2 | −1, {1 | −1}}
territory count +2		 N When a real board falls apart
..LxxR.xx 1 | −1
territory count +1		 N Amazons, and when a position becomes a sum, When a real board falls apart
..RLxx.x. 2 | 0
territory count +1		 N When a real board falls apart
..RxxL.xx 1 | −1
territory count −1		 N Amazons, and when a position becomes a sum, When a real board falls apart
.x..xxLRx 1 | −1
territory count +1		 N When a real board falls apart
1×2 — L. 1 L Amazons on one line
1×2 — LR 0 P Amazons on one line
1×3 — .L. 2 L Amazons on one line
1×3 — L.. 2 L Amazons on one line
1×3 — L.R ∗ N Amazons on one line
1×4 — .L.R 1∗ L Amazons on one line
1×4 — L... 3 L Amazons on one line
1×4 — L..R 1 | −1 N Amazons on one line
1×4 — L.xR 1 L Amazons on one line
1×4 — Lx.R −1 R Amazons on one line
1×5 — ..L.. 4 L Amazons on one line
1×5 — L.... 4 L Amazons on one line
1×5 — L...R 2 | −2 N Amazons on one line
1×5 — L.x.R 0 P Amazons on one line
1×6 — .L..R. 1 | −1 N Amazons on one line
1×6 — L....R 3 | −3 N Amazons on one line
1×6 — L..x.R 1 L Amazons on one line
1×6 — L.x..R −1 R Amazons on one line
1×7 — L.....R 4 | −4 N Amazons on one line
1×7 — L.x.x.R 0 P Amazons on one line, When a real board falls apart
1×7 — L.x.x.R, region 1 1 L Amazons on one line, When a real board falls apart
1×7 — L.x.x.R, region 2 0 P Amazons on one line, When a real board falls apart
1×7 — L.x.x.R, region 3 −1 R Amazons on one line, When a real board falls apart
1×8 — ..L..R.. 1 | −1 N Amazons on one line
2×4 — L......R {3, {4 | 0, {1 | 0, ∗}} | −3, {0, {0, ∗ | −1} | −4}} N Amazons, and when a position becomes a sum, Amazons on one line
2×5 — L...R.x.x. 5 | −5 N Amazons, and when a position becomes a sum
2×5 — L.xR.xxx.L 1 L When a real board falls apart
2×5 — L.xR.xxx.L, region 1 1 L When a real board falls apart
2×5 — L.xR.xxx.L, region 2 0 P When a real board falls apart
2×6 — L.xx.RxxxxxL 1∗ L Amazons, and when a position becomes a sum
2×6 — L.xx.RxxxxxL, region 1 1 L Amazons, and when a position becomes a sum
2×6 — L.xx.RxxxxxL, region 2 ∗ N Amazons, and when a position becomes a sum
2×7 — L.xx.xR.xx.x.. −2 R When a real board falls apart
2×7 — L.xx.xR.xx.x.., region 1 2 L When a real board falls apart
2×7 — L.xx.xR.xx.x.., region 2 −4 R When a real board falls apart
3×3 — L.......R {{5 | {1∗ | 0, {1/2 | 0}}, {2 | −1/4}} | {{0, {0 | −1/2} | −1∗}, {1/4 | −2} | −5}} N Amazons, and when a position becomes a sum
3×3 — L.x..x... 6 L A region one player owns
3×4 — L..R..x..x.. {7 | −3, {−2 | −5}} N Amazons, and when a position becomes a sum
3×4 — L.xRxxxx.L.R 2∗ L A wall an amazon can walk through, When a real board falls apart
3×4 — L.xRxxxx.L.R, region 1 1 L A wall an amazon can walk through, When a real board falls apart
3×4 — L.xRxxxx.L.R, region 2 0 P A wall an amazon can walk through, When a real board falls apart
3×4 — L.xRxxxx.L.R, region 3 1∗ L A wall an amazon can walk through, When a real board falls apart
3×5 — L.x.R..x....xxx 2 L Amazons, and when a position becomes a sum, The board falls apart, and the arithmetic changes, When a real board falls apart
3×5 — L.x.R..x....xxx, region 1 5 L Amazons, and when a position becomes a sum, The board falls apart, and the arithmetic changes, When a real board falls apart
3×5 — L.x.R..x....xxx, region 2 −3 R Amazons, and when a position becomes a sum, The board falls apart, and the arithmetic changes, When a real board falls apart
3×5 — L.x.R.x.x....xx {3 | {1, {1 | −1} | −2, {−1 | −4}}} L When a real board falls apart
3×5 — L.x.RxxxxxL.x.R 0 P Amazons, and when a position becomes a sum
3×5 — L.x.RxxxxxL.x.R, region 1 1 L Amazons, and when a position becomes a sum
3×5 — L.x.RxxxxxL.x.R, region 2 −1 R Amazons, and when a position becomes a sum
3×5 — L.x.RxxxxxL.x.R, region 3 1 L Amazons, and when a position becomes a sum
3×5 — L.x.RxxxxxL.x.R, region 4 −1 R Amazons, and when a position becomes a sum

Bidding

12 positions

PositionWorthOutcomeDrawn in
↑ — Richman value 1/2 L Nobody has to move
↓ — Richman value 1/2 R Nobody has to move
−1 — Richman value 1/4 R Nobody has to move
−2 — Richman value 1/8 R Nobody has to move
∗ — Richman value 1/2 N Nobody has to move
0 — Richman value 1/2 P Nobody has to move
1 — Richman value 3/4 L Nobody has to move
1 | −1 — Richman value 1/2 N Nobody has to move
1/2 — Richman value 5/8 L Nobody has to move
2 — Richman value 7/8 L Nobody has to move
2 | 0 — Richman value 11/16 N Nobody has to move
3 — Richman value 15/16 L Nobody has to move

Chomp

6 positions

PositionWorthOutcomeDrawn in
1 × 2 bar N
1 winning first move		 N The first theorem, and the winner it declines to name, The theorem that names a winner and no move
2 × 1 bar N
1 winning first move		 N The first theorem, and the winner it declines to name, The theorem that names a winner and no move
3 × 1 bar N
1 winning first move		 N The first theorem, and the winner it declines to name, The theorem that names a winner and no move
4 × 1 bar N
1 winning first move		 N The first theorem, and the winner it declines to name, The theorem that names a winner and no move
5 × 1 bar N
1 winning first move		 N The first theorem, and the winner it declines to name, The theorem that names a winner and no move
6 × 1 bar N
1 winning first move		 N The first theorem, and the winner it declines to name, The theorem that names a winner and no move

Clobber

37 positions

PositionWorthOutcomeDrawn in
1×1 x 0 P A game where nobody can be ahead in moves
1×2 xo ∗ N The class where nobody runs out first, A game where nobody can be ahead in moves, One row of Clobber, When the ups add
1×3 oxo ∗ N A game where nobody can be ahead in moves
1×3 xoo ↓ R The class where nobody runs out first, A game where nobody can be ahead in moves
1×3 xox ∗ N The class where nobody runs out first, A game where nobody can be ahead in moves
1×3 xxo ↑ L The class where nobody runs out first, A game where nobody can be ahead in moves, When the ups add
1×3 xxx 0 P The class where nobody runs out first
1×4 oxoo ↑∗ N One row of Clobber
1×4 xooo 2·↓∗ R One row of Clobber
1×4 xoxo {↑, ∗ | ↓, ∗} N The class where nobody runs out first, A game where nobody can be ahead in moves, One row of Clobber, When the bracket decides, When the ups add
1×4 xxoo 0 P One row of Clobber
1×4 xxox ↓∗ N One row of Clobber
1×4 xxxo 2·↑∗ L One row of Clobber
1×4 xxxx 0 P One row of Clobber
1×5 x.o.x 0 P A game where nobody can be ahead in moves
1×5 xo.xo 0 P A game where nobody can be ahead in moves
1×5 xoxox {∗ | ↓} R The class where nobody runs out first, One row of Clobber
1×5 xx.oo 0 P One row of Clobber
1×5 xxoxo ∗ N When the bracket decides
1×6 xoxoxo 0 P One row of Clobber, When the bracket decides
1×6 xoxxox 0 P One row of Clobber
1×7 xox.xox 0 P One row of Clobber
1×7 xoxo.xo {0, ↑∗ | 0, ↓∗} N One row of Clobber
1×7 xoxoxox {{↑, ∗ | ↓, ∗}, ↓ | 0} R One row of Clobber
1×8 xo.xo.xo ∗ N A game where nobody can be ahead in moves
1×8 xoxo.oxo {0, ↑∗ | 0, ↓∗} N One row of Clobber
1×8 xxooxxoo 0 P One row of Clobber
1×8 xxxxoooo 0 P One row of Clobber
2×2 xoox ∗ N The class where nobody runs out first, A game where nobody can be ahead in moves
2×2 xoxo 0 P The class where nobody runs out first, A game where nobody can be ahead in moves, When the ups add
2×3 xo.oxo {↑, ∗ | 2·↓∗} N The class where nobody runs out first
2×3 xoxoxo 0 P The class where nobody runs out first, A game where nobody can be ahead in moves, Three different claims are all called solved
2×4 xoxoxoxo 0 P The class where nobody runs out first, A game where nobody can be ahead in moves
3×3 xoxoxoxox ∗ N The class where nobody runs out first, A game where nobody can be ahead in moves
Clobber 1×3 xxo ↑ L When the ups add
Clobber 2×4 0
55 positions walked for a value with 1 node		 P A game where nobody can be ahead in moves, Maundy Cake
Clobber 3×3 ∗
91 positions walked for a value with 2 nodes		 N A game where nobody can be ahead in moves, The heap is not the position, A position reached eleven ways is one position

Col

11 positions

PositionWorthOutcomeDrawn in
a bowtie 0 P One rule makes it cold, the other hot
a bowtie from L.... −1 R One board, two rules
a path of four 0 P One board, two rules, Every group must keep breathing
a path of four from .L.. −3/2 R One rule makes it cold, the other hot
a path of three 0 P One board, two rules, One rule makes it cold, the other hot
a path of three from .L. −2 R One board, two rules
a path of three from L.. −1/2 R One board, two rules
a star ∗ N One board, two rules
a star from ..LR ∗ N One rule makes it cold, the other hot
a star from L... −3 R One rule makes it cold, the other hot
a triangle 0 P One board, two rules

Cram

11 positions

PositionWorthOutcomeDrawn in
1×1 0
the pairing strategy does not reach it		 P Cram, Looking for the symmetry
1×5 0
the pairing strategy does not reach it		 P Cram, Looking for the symmetry, The strategy that is a symmetry
2×2 0
settled by the pairing strategy		 P Cram, Looking for the symmetry
2×3 ∗1 N Cram, The strategy that is a symmetry
2×4 0
settled by the pairing strategy		 P Cram, Looking for the symmetry, The strategy that is a symmetry
2×5 ∗1
the pairing strategy breaks here		 N The strategy that is a symmetry
3×3 0
the pairing strategy breaks here		 P Cram, The strategy that is a symmetry
3×4 ∗1
the pairing strategy breaks here		 N Cram
3×4 with 1 domino placed 0
the pairing strategy holds over 11 positions		 P Cram
4×4 0
the pairing strategy holds over 60 positions		 P Cram, The strategy that is a symmetry, The symmetry one move away
5×5 0
the pairing strategy breaks here		 P Cram

Cutcake

65 positions

PositionWorthOutcomeDrawn in
1×1 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
1×2 1 L Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
1×3 2 L Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
1×4 3 L Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
1×5 4 L Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
1×6 5 L Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
1×7 6 L Maundy Cake
1×8 7 L Maundy Cake
2×1 −1 R Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
2×2 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
2×3 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
2×4 1 L Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
2×5 1 L Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
2×6 2 L Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
2×7 2 L Maundy Cake
2×8 3 L Maundy Cake
3×1 −2 R Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
3×2 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
3×3 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
3×4 1 L Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
3×5 1 L Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
3×6 2 L Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
3×7 2 L Maundy Cake
3×8 3 L Maundy Cake
4×1 −3 R Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
4×2 −1 R Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
4×3 −1 R Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
4×4 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
4×5 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
4×6 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
4×7 0 P Maundy Cake
4×8 1 L Maundy Cake
5×1 −4 R Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
5×2 −1 R Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
5×3 −1 R Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
5×4 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
5×5 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
5×6 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
5×7 0 P Maundy Cake
5×8 1 L Maundy Cake
6×1 −5 R Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
6×2 −2 R Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
6×3 −2 R Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
6×4 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
6×5 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
6×6 0 P Cutcake, where every value is a whole number, Maundy Cake, The game with the shortest rule is the hard one, The obvious cut is the wrong one
6×7 0 P Maundy Cake
6×8 1 L Maundy Cake
7×1 −6 R Maundy Cake
7×2 −2 R Maundy Cake
7×3 −2 R Maundy Cake
7×4 0 P Maundy Cake
7×5 0 P Maundy Cake
7×6 0 P Maundy Cake
7×7 0 P Maundy Cake
7×8 1 L Maundy Cake
8×1 −7 R Maundy Cake
8×2 −3 R Maundy Cake
8×3 −3 R Maundy Cake
8×4 −1 R Maundy Cake
8×5 −1 R Maundy Cake
8×6 −1 R Maundy Cake
8×7 −1 R Maundy Cake
8×8 0 P Maundy Cake
Cutcake 5×5 0
2903 positions walked for a value with 1 node		 P Maundy Cake

Dawson's chess ·137

14 positions

PositionWorthOutcomeDrawn in
Dawson's chess ·137, a heap of 1 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild, The genus of a sum, What a tame heap may be replaced by
Dawson's chess ·137, a heap of 10 genus 3^31
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
Dawson's chess ·137, a heap of 11 genus 2^0520
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild
Dawson's chess ·137, a heap of 12 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
Dawson's chess ·137, a heap of 13 genus 4^146
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
Dawson's chess ·137, a heap of 14 genus 0^120
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
Dawson's chess ·137, a heap of 2 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild, The genus of a sum, What a tame heap may be replaced by
Dawson's chess ·137, a heap of 3 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, The genus of a sum, What a tame heap may be replaced by
Dawson's chess ·137, a heap of 4 genus 0^120
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, The genus of a sum, What a tame heap may be replaced by
Dawson's chess ·137, a heap of 5 genus 3^31
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, The genus of a sum
Dawson's chess ·137, a heap of 6 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild, The genus of a sum
Dawson's chess ·137, a heap of 7 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild, The genus of a sum
Dawson's chess ·137, a heap of 8 genus 0^120
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, The genus of a sum
Dawson's chess ·137, a heap of 9 genus 3^1431
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, The genus of a sum

Domineering

62 positions

PositionWorthOutcomeDrawn in
0,0 0,1 0,2 1,0 1,2 0
packs +1 and is worth 0		 P Counting the moves each side has, The moves a player can be talked out of
0,0 0,1 1,0 2,0 2,1 0
packs −1 and is worth 0		 P Counting the moves each side has, The moves a player can be talked out of
0,0 0,1 1,1 2,1 2,2 0
packs −1 and is worth 0		 P Counting the moves each side has, The moves a player can be talked out of
1 × 1 0 P The operator chosen for one game
1 × 2 −1 R The operator chosen for one game
1 × 3 −1 R The operator chosen for one game
1×1 0 P Where the impartial theory stops
1×2 −1 R A board that is a sum of its regions, One board, two rules, Domineering, Worth nothing, and worth fighting for, The values of every small board, Which option the reduction keeps, Which shapes are worth fighting over
1×2 board −1
cools by 1 to −1		 R Below zero
1×2 Domineering −1 R What can be struck out
1×3 −1 R Domineering, The values of every small board
1×3 board −1
cools by 1 to −1		 R Cooling by exactly one
1×4 −2 R The values of every small board, What can be struck out
1×4 cut at column 1 −1
the sum of the two halves		 R Independence is a claim
1×4 whole −2
evaluated as one board		 R Independence is a claim
1×5 −2 R The values of every small board
1×6 −3 R The values of every small board
1×7 −3 R The values of every small board
2 × 1 1 L The operator chosen for one game
2×1 1 L Domineering
2×2 1 | −1 N A board that is a sum of its regions, One board, two rules, Cooling by exactly one, Domineering, Worth nothing, and worth fighting for, The values of every small board, Tiny, miny, and the sizes below every size, What can be struck out, What counts as the same position, and what that is worth, Which option the reduction keeps, Which shapes are worth fighting over
2×2 board 1 | −1
cools by 1 to ∗		 N Below zero, Cooling by exactly one, The operator chosen for one game
2×2 Domineering 1 | −1 N What can be struck out
2×3 2 | −1/2 N A board that is a sum of its regions, "Left wins" has no short proof, One board, two rules, Cooling by exactly one, Domineering, Turn the board through a right angle, Worth nothing, and worth fighting for, The values of every small board, Tiny, miny, and the sizes below every size, What can be struck out, What counts as the same position, and what that is worth, Where the impartial theory stops, Which option the reduction keeps, Which shapes are worth fighting over
2×3 board 2 | −1/2
cools by 1 to 1 | 1/2		 N Cooling by exactly one, The operator chosen for one game
2×3 Domineering 2 | −1/2 N What can be struck out
2×3 less two squares ∗ N Where the impartial theory stops
2×4 {{2 | 0} | 0} R Cooling by exactly one, Turn the board through a right angle, The values of every small board, Tiny, miny, and the sizes below every size, Which shapes are worth fighting over
2×4 board {{2 | 0} | 0}
cools by 1 to 0		 R Below zero, Cooling by exactly one, The operator chosen for one game
2×4 cut at column 2 0
the sum of the two halves		 P Independence is a claim, The sum is the object
2×4 whole {{2 | 0} | 0}
evaluated as one board		 R Independence is a claim, The sum is the object
2×5 1/2
winning strategy: 18 positions, 9 prescribed moves		 L "Left wins" has no short proof, The values of every small board
2×5 cut at column 2 {{3 | 1} | {1/2 | −3/2}}
the sum of the two halves		 L Finding the parts, How wrong a nearly-independent split is, Independence is a claim, When the regions add
2×5 cut at column 3 {{3 | 1} | {1/2 | −3/2}}
the sum of the two halves		 L Independence is a claim
2×5 whole 1/2
evaluated as one board		 L Finding the parts, How wrong a nearly-independent split is, Independence is a claim, When the regions add
2×6 {{{3 | 1} | 1} | −1} N The values of every small board
2×7 3/2 | −1/2 N The values of every small board
2×8 {{{{4 | 2} | 2} | 0} | {−1/2 | −2}} R The values of every small board
2×9 {{5/2 | 1/2} | {0 | −3/2}} L The values of every small board
3 × 1 1 L The operator chosen for one game
3×2 1/2 | −2
winning strategy: 4 positions, 2 prescribed moves		 N "Left wins" has no short proof, Cooling by exactly one, Turn the board through a right angle
3×3 1 | −1 N A board that is a sum of its regions, "Left wins" has no short proof, One board, two rules, Cooling by exactly one, Cram, Domineering, Worth nothing, and worth fighting for, The values of every small board, Tiny, miny, and the sizes below every size, What counts as the same position, and what that is worth, Which option the reduction keeps, Which shapes are worth fighting over
3×3 after Left's move 2
an option the reduction keeps		 L The margin a count needs, Which option the reduction keeps
3×3 board 1 | −1
cools by 1 to ∗		 N Below zero, Cooling by exactly one, The operator chosen for one game
3×3 less four squares 1∗ L Where the impartial theory stops
3×4 −3/2 R A board that is a sum of its regions, "Left wins" has no short proof, Domineering, A position reached eleven ways is one position, The values of every small board, Which shapes are worth fighting over
3×4 board −3/2
cools by 1 to −3/2		 R Below zero
3×4 cut at column 1 2 | 0
the sum of the two halves		 N How wrong a nearly-independent split is, Independence is a claim
3×4 whole −3/2
evaluated as one board		 R How wrong a nearly-independent split is, Independence is a claim
3×5 −1 R The values of every small board, Which shapes are worth fighting over
3×6 −1 | −7/2 R The values of every small board
3×7 −3/4 | −3 R The values of every small board
4×2 {0 | {0 | −2}} L Turn the board through a right angle
4×3 cut at column 1 {2 | {2 | 0}}
the sum of the two halves		 L Independence is a claim
4×3 whole 3/2
evaluated as one board		 L Independence is a claim
4×4 {0, {{2 | 0}, {2 | {2 | 0}} | {2 | 0}, {{2 | 0} | 0}} | 0, {{0 | {0 | −2}}, {0 | −2} | {0 | −2}, {{0 | −2} | −2}}} N The values of every small board, Which shapes are worth fighting over
4×5 1 L The values of every small board
a strip of 1 0
one orientation fits and the value is the packing count		 P Counting the moves each side has, The moves a player can be talked out of
a strip of 2 −1
one orientation fits and the value is the packing count		 R Counting the moves each side has, The moves a player can be talked out of
a strip of 3 −1
one orientation fits and the value is the packing count		 R Counting the moves each side has, The moves a player can be talked out of
Domineering 3×3 1 | −1
18 positions walked for a value with 4 nodes		 N Knowing who wins, and knowing what it is worth, Maundy Cake, The heap is not the position, A position reached eleven ways is one position
Domineering 4×4 {0, {{2 | 0}, {2 | {2 | 0}} | {2 | 0}, {{2 | 0} | 0}} | 0, {{0 | {0 | −2}}, {0 | −2} | {0 | −2}, {{0 | −2} | −2}}}
562 positions walked for a value with 14 nodes		 N Knowing who wins, and knowing what it is worth, A position reached eleven ways is one position

Domineering region

6 positions

PositionWorthOutcomeDrawn in
4 squares, 0,0 0,1 1,0 1,1 1 | −1
drawn beside its own value		 N Which shapes are worth fighting over
5 squares, 0,0 0,1 0,2 1,0 1,1 1 | −1
drawn beside its own value		 N Which shapes are worth fighting over
6 squares, 0,0 0,1 0,2 1,0 1,1 1,2 2 | −1/2
drawn beside its own value		 N Which shapes are worth fighting over
6 squares, 0,0 0,1 1,0 1,1 1,2 2,2 2 | −1/2
drawn beside its own value		 N Which shapes are worth fighting over
6 squares, 0,0 0,1 1,0 1,1 2,0 2,1 1/2 | −2
drawn beside its own value		 N Which shapes are worth fighting over
6 squares, 0,0 0,1 1,0 1,1 2,1 2,2 1/2 | −2
drawn beside its own value		 N Which shapes are worth fighting over

Elephants and Rhinos

21 positions

PositionWorthOutcomeDrawn in
T...F ∗ N The same strip without the jump
T..TF 2 L The same strip without the jump, The strip where every number is a whole one
T.F ∗ N The same strip without the jump
T.F. ∗ N The same strip without the jump
T.FF 0 | −1 N The same strip without the jump
T.FT.F 0 P The same strip without the jump
T.FTT.F {1∗ | ∗} L The same strip without the jump
T.T.F 2 | 1 L The strip where every number is a whole one
T.TF 1 L The same strip without the jump
T.TF.F 0 P The same strip without the jump
T.TFF 1 L The same strip without the jump
TF.. 0 P The same strip without the jump
TF.F −1 R The same strip without the jump
TT..F 1 L The strip where every number is a whole one
TT..FF 0 P The same strip without the jump, The strip where every number is a whole one
TT.F 1 | 0 N The same strip without the jump, The strip where every number is a whole one
TT.F.FT 0 | −2 N The same strip without the jump
TT.FT.F {1∗ | ∗} L The strip where every number is a whole one
TTF..F −2 R The same strip without the jump
TTF.F −1 R The same strip without the jump
TTT.F 2 | 0 N The same strip without the jump

End-Nim

25 positions

PositionWorthOutcomeDrawn in
1 ∗ N Taking from the ends
1, 1 0 P Taking from the ends, Where the nimbers run out
1, 1, 1 ∗ N Taking from the ends
1, 2 {∗2 | 0} R Taking from the ends
1, 2, 1 ∗2 N Taking from the ends, Where the nimbers run out
1, 2, 2 ↓∗ N Taking from the ends
1, 2, 2, 1 0 P Taking from the ends, Where the nimbers run out
1, 2, 3 {{{∗3 | 0}, ∗3 | 0} | 0} R Taking from the ends, Where the nimbers run out
1,2,1,3 0
worth 0, and so is 3, 1, 2, 1		 P The rows that are their own mirror, Where the nimbers run out
1,3,1,4 0
worth 0, and so is 4, 1, 3, 1		 P The rows that are their own mirror, Where the nimbers run out
1,3,2,4 0
worth 0, and so is 4, 2, 3, 1		 P The rows that are their own mirror, Where the nimbers run out
2 ∗2 N Taking from the ends
2, 1 {0 | ∗2} L Taking from the ends
2, 1, 2 0 P Taking from the ends, Where the nimbers run out
2, 2 0 P Taking from the ends
2, 2, 2 ∗ N Taking from the ends, Where the nimbers run out
2,2,1,4 0
worth 0, and so is 4, 1, 2, 2		 P The rows that are their own mirror, Where the nimbers run out
3 ∗3 N Taking from the ends
3, 1 {0 | ∗3} L Taking from the ends, Where the nimbers run out
3, 2, 3 0 P Taking from the ends, Where the nimbers run out
3,1,2,1 0
not a palindrome, and worth a nimber		 P Where the nimbers run out
4 ∗4 N Taking from the ends
4, 4 0 P Taking from the ends, Where the nimbers run out
4,1,2,2 0
not a palindrome, and worth a nimber		 P Where the nimbers run out
5 ∗5 N Taking from the ends

Fibonacci Nim

4 positions

PositionWorthOutcomeDrawn in
a heap of 3 first player loses P The family the Fibonacci numbers belong to, The heap is not the position, What restores the theorem
a heap of 4 first player wins N The family the Fibonacci numbers belong to, The heap is not the position, What restores the theorem
a heap of 8 first player loses P The family the Fibonacci numbers belong to, The heap is not the position, What restores the theorem
a heap of 9 first player wins N The family the Fibonacci numbers belong to, The heap is not the position, What restores the theorem

Form

2 positions

PositionWorthOutcomeDrawn in
{{0 | ∗2} | 0} ↑∗ N A reduction that reads a graph, The reduction that always shrinks, The reduction that puts options back
{0, ↑ | 0, ∗2, ↓} ∗ N How much a list of options can lose, How wide a form can get

Generalized Geography

6 positions

PositionWorthOutcomeDrawn in
14-node reduction of a 3-variable formula —
the formula is true, and the game agrees		 N A puzzle asks once, a game asks alternately
16-node reduction of a 3-variable formula —
the formula is true, and the game agrees		 N A puzzle asks once, a game asks alternately, It ends, and nothing says when
17-node reduction of a 3-variable formula —
the formula is false, and the game agrees		 P A puzzle asks once, a game asks alternately
26-node reduction of a 5-variable formula —
the formula is true, and the game agrees		 N A puzzle asks once, a game asks alternately
6-node reduction of a 1-variable formula —
the formula is true, and the game agrees		 N A puzzle asks once, a game asks alternately
7-node reduction of a 1-variable formula —
the formula is false, and the game agrees		 P A puzzle asks once, a game asks alternately

Geography

12 positions

PositionWorthOutcomeDrawn in
undirected on 5 vertices and 4 edges, starting at a ∗0
2 of 3 maximum matchings cover it		 P A token on a graph
undirected on 5 vertices and 4 edges, starting at b ∗1
3 of 3 maximum matchings cover it		 N A token on a graph
undirected on 5 vertices and 4 edges, starting at c ∗0
2 of 3 maximum matchings cover it		 P A token on a graph
undirected on 5 vertices and 4 edges, starting at d ∗1
3 of 3 maximum matchings cover it		 N A token on a graph
undirected on 6 vertices and 5 edges, starting at a ∗0
2 of 4 maximum matchings cover it		 P A token on a graph
undirected on 6 vertices and 5 edges, starting at b ∗2
4 of 4 maximum matchings cover it		 N A token on a graph
undirected on 6 vertices and 5 edges, starting at c ∗0
2 of 4 maximum matchings cover it		 P A token on a graph
undirected on 6 vertices and 5 edges, starting at d ∗2
4 of 4 maximum matchings cover it		 N A token on a graph
undirected on 6 vertices and 8 edges, starting at a ∗1
3 of 3 maximum matchings cover it		 N Hard, proved
undirected on 6 vertices and 8 edges, starting at b ∗1
3 of 3 maximum matchings cover it		 N Hard, proved
undirected on 6 vertices and 8 edges, starting at c ∗1
3 of 3 maximum matchings cover it		 N Hard, proved
undirected on 6 vertices and 8 edges, starting at d ∗2
3 of 3 maximum matchings cover it		 N Hard, proved

Green Hackenbush

7 positions

PositionWorthOutcomeDrawn in
a square with a roof ∗0 P Squash every loop to a point
a three-by-three lattice on the ground ∗0 P Squash every loop to a point, Nim is easy, in binary
a tree with two branches ∗4 N Squash every loop to a point, When the nested sum only sees the value
a triangle on a stalk ∗2 N A tree is still a number, Squash every loop to a point
a triangle standing on the ground ∗1 N Squash every loop to a point
a triangle with a tail ∗0 P A green edge on a blue one
two loops on one stalk ∗2 N A winning strategy that is a spanning tree

Hackenbush

44 positions

PositionWorthOutcomeDrawn in
E ∗ N How many ups, Squash every loop to a point, Hackenbush is a numeral, Three ways to add the same games, Outcomes do not add, The other sum, the one that nests, The sum is the object
EE ∗2 N Squash every loop to a point, Hackenbush is a numeral, The other sum, the one that nests
EEE ∗3 N Hackenbush is a numeral, The other sum, the one that nests
EEEE ∗4 N The other sum, the one that nests
EEL {0, ∗, ∗2 | 0, ∗} N A green edge on a blue one
EL ↑∗ N A green edge on a blue one, How many ups, Squash every loop to a point, The other sum, the one that nests
ELL {0, ↑∗ | 0} N A green edge on a blue one, How many ups
ELR {0, ∗ | 0, ↑∗} N A green edge on a blue one
ER ↓∗ N A green edge on a blue one
green EE ∗2 N When the ups add
L 1 L A green edge on a blue one, Squash every loop to a point, Hackenbush is a numeral, Nothing worth fighting over, The numbers came out of the game, The other sum, the one that nests, The reading that survives too much, The simplicity rule, The values nobody's game produces, What the colon respects
LE 1∗ L A green edge on a blue one, How many ups, Squash every loop to a point, Hackenbush is a numeral, The numbers came out of the game, The other sum, the one that nests
LEE 1∗2 L A green edge on a blue one
LEEE 1∗3 L A green edge on a blue one
LEL 1↑∗ L A green edge on a blue one
LERL {1, 1↓∗ | 1, 1∗} L A green edge on a blue one
LL 2 L Squash every loop to a point, Hackenbush is a numeral, Nothing worth fighting over, Three ways to add the same games, Outcomes do not add, The numbers came out of the game, The simplicity rule, The sum is the object
LL + R + LRR + E 5/4∗ L Three ways to add the same games, Outcomes do not add, The sum is the object
LLE 2∗ L A green edge on a blue one, The numbers came out of the game
LLL 3 L Hackenbush is a numeral, The numbers came out of the game
LLLL 4 L Hackenbush is a numeral, The numbers came out of the game
LLR 3/2 L The numbers came out of the game
LLRL 7/4 L Hackenbush is a numeral
LLRL + RRRL −3/4 R Hackenbush is a numeral
LR 1/2 L Canonical form, Comparing positions, Squash every loop to a point, Hackenbush is a numeral, Nobody comes back, Nothing worth fighting over, The numbers came out of the game, The other sum, the one that nests, The reading that survives too much, The simplicity rule, The sum is the object, The values nobody's game produces, What the colon respects
LR + RL 0 P The sum is the object
LRE 1/2∗ L A green edge on a blue one, The numbers came out of the game
LRL 3/4 L Squash every loop to a point, Hackenbush is a numeral, Nothing worth fighting over, The numbers came out of the game, The other sum, the one that nests, The reading that survives too much, The simplicity rule, The values nobody's game produces, What the colon respects, When the nested sum only sees the value
LRLL 7/8 L What the colon respects
LRLR 5/8 L Hackenbush is a numeral, The numbers came out of the game, The other sum, the one that nests
LRLRL 11/16 L The numbers came out of the game, The other sum, the one that nests
LRLRLR 21/32 L The numbers came out of the game
LRR 1/4 L Squash every loop to a point, Hackenbush is a numeral, Nothing worth fighting over, Three ways to add the same games, Outcomes do not add, The numbers came out of the game, The other sum, the one that nests, The reading that survives too much, The simplicity rule, The sum is the object, The values nobody's game produces
LRRL 3/8 L Squash every loop to a point, Hackenbush is a numeral, Nothing worth fighting over, The reading that survives too much, The simplicity rule, The values nobody's game produces
LRRR 1/8 L Hackenbush is a numeral, The numbers came out of the game, The other sum, the one that nests
LRRRR 1/16 L The numbers came out of the game, The other sum, the one that nests
R −1 R Three ways to add the same games, Outcomes do not add, The other sum, the one that nests, The sum is the object
RE −1∗ R A green edge on a blue one
RL −1/2 R The other sum, the one that nests, The sum is the object
RLR −3/4 R Hackenbush is a numeral
RR −2 R The other sum, the one that nests
RRE −2∗ R A green edge on a blue one
RRLR −7/4 R Hackenbush is a numeral
RRRL −5/2 R Hackenbush is a numeral

Hackenbush tree

11 positions

PositionWorthOutcomeDrawn in
above the trunk 0 P A tree is still a number
cherry 1 L A tree is still a number, Where the numeral stops
fan 1/8 L A tree is still a number, Where the numeral stops
ladder 3/2 L A tree is still a number, Where the numeral stops
LR 1/2 L A tree is still a number
LRL 3/4 L A tree is still a number, Where the numeral stops
red-cherry −1 R A tree is still a number, Where the numeral stops
RL −1/2 R A tree is still a number
tree 1 −1/4 R A tree is still a number
two-trunks 0 P A tree is still a number, Where the numeral stops
whole tree 1 L A tree is still a number

Hexadecimal game

4 positions

PositionWorthOutcomeDrawn in
·8f, a heap of 0 0 P A period with a constant added, A code that climbs by three
·8f, a heap of 1 0 P A period with a constant added, A code that climbs by three
·8f, a heap of 2 ∗1 N A period with a constant added, A code that climbs by three
·8f, a heap of 3 ∗1 N A period with a constant added, A code that climbs by three

Impartial

3 positions

PositionWorthOutcomeDrawn in
a heap of 4, taking 1, 3, 4 ∗2
equal to exactly one nimber in the range drawn		 N Where the impartial theory stops
a heap of 5, taking 1, 3, 4 ∗3
equal to exactly one nimber in the range drawn		 N Where the impartial theory stops
a heap of 7, taking 1, 3, 4 0
equal to exactly one nimber in the range drawn		 P Where the impartial theory stops

Kayles ·77

14 positions

PositionWorthOutcomeDrawn in
Kayles ·77, a heap of 1 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P A function with no formula, Closing the wild side, Tame and wild, The genus of a sum, The rule the symbols follow, What a tame heap may be replaced by
Kayles ·77, a heap of 10 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Closing the wild side, Tame and wild, The genus of a sum
Kayles ·77, a heap of 11 genus 6^46
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Closing the wild side, Tame and wild, The genus of a sum
Kayles ·77, a heap of 12 genus 4^046
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Closing the wild side, Tame and wild, The genus of a sum
Kayles ·77, a heap of 13 genus 1^13
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
Kayles ·77, a heap of 14 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
Kayles ·77, a heap of 2 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N A function with no formula, Closing the wild side, Tame and wild, The genus of a sum, The rule the symbols follow, What a tame heap may be replaced by
Kayles ·77, a heap of 3 genus 3^31
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N A function with no formula, Closing the wild side, Tame and wild, The genus of a sum, The rule the symbols follow, What a tame heap may be replaced by
Kayles ·77, a heap of 4 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P A function with no formula, Closing the wild side, Tame and wild, The genus of a sum, The rule the symbols follow, What a tame heap may be replaced by
Kayles ·77, a heap of 5 genus 4^146
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N A function with no formula, Closing the wild side, Tame and wild, The genus of a sum, The rule the symbols follow, What a tame heap may be replaced by
Kayles ·77, a heap of 6 genus 3^31
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N A function with no formula, Closing the wild side, Tame and wild, The genus of a sum, The rule the symbols follow, What a tame heap may be replaced by
Kayles ·77, a heap of 7 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N A function with no formula, Closing the wild side, Tame and wild, The genus of a sum, The rule the symbols follow, What a tame heap may be replaced by
Kayles ·77, a heap of 8 genus 1^13
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N A function with no formula, Closing the wild side, Tame and wild, The genus of a sum, The rule the symbols follow, What a tame heap may be replaced by
Kayles ·77, a heap of 9 genus 4^046
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P A function with no formula, Closing the wild side, Tame and wild, The genus of a sum

Kōnane

51 positions

PositionWorthOutcomeDrawn in
.....ox 1
1 moves for Left, 0 for Right		 L A game older than the theory
....xo. ∗
1 moves for Left, 1 for Right		 N A game older than the theory
...ox.x 1/2
1 moves for Left, 1 for Right		 L A game older than the theory, The second dimension is not the deep end
..x.xo. ↓∗
1 moves for Left, 2 for Right		 N A game older than the theory
..x/oxo/xox 0
after Black lifts rank 1 file 1		 P The two moves that are not captures
..xo.o. ↑∗
2 moves for Left, 1 for Right		 N A game older than the theory
..xo/oxox/xoxo 0
after Black lifts rank 1 file 1		 P The two moves that are not captures
..xox/oxoxo/xoxox 0
after Black lifts rank 1 file 1		 P The two moves that are not captures
.ox.x.x 1/4
1 moves for Left, 2 for Right		 L A game older than the theory, The second dimension is not the deep end
.oxo.xo 1/2
2 moves for Left, 1 for Right		 L A game older than the theory
.x...xo −1
pieces sum to −1		 R Independence is a claim
.x..xo. ∗
pieces sum to 1		 N A game older than the theory, Independence is a claim
.xo.xo. 1 | −1
2 moves for Left, 2 for Right		 N A game older than the theory
.xoo.ox 1/2
1 moves for Left, 1 for Right		 L A game older than the theory
.xoxox. −2
0 moves for Left, 2 for Right		 R A game older than the theory, The two moves that are not captures
oxo.oxo.oxo 0
playable: the reader moves first and loses		 P A game older than the theory
oxox.xoxo 0
0 moves for Left, 2 for Right		 P A game older than the theory, The two moves that are not captures
x...x.x 0
0 moves for Left, 0 for Right		 P A game older than the theory, The second dimension is not the deep end
x../oxo/xox 0
after Black lifts rank 1 file 3		 P The two moves that are not captures
x.ox 1
1 moves for Left, 0 for Right		 L A game older than the theory
x.x.xo. 1/4
pieces sum to 1/4		 L A game older than the theory, Independence is a claim
x.x/.o./x.x 0
3 × 3, 0 moves for Left and 0 for Right		 P The second dimension is not the deep end
x.xo..o 1/2
1 moves for Left, 1 for Right		 L A game older than the theory
xo. 1
1 moves for Left, 0 for Right		 L A game older than the theory
xo../oxox/xoxo 0
after Black lifts rank 1 file 3		 P The two moves that are not captures
xo.ox/ox.xo/xoxox 0
after Black lifts rank 1 file 3		 P The two moves that are not captures
xox../oxoxo/xoxox 0
after Black lifts rank 1 file 5		 P The two moves that are not captures
xox/o../xox 0
after Black lifts rank 2 file 2		 P The two moves that are not captures
xox/o.o/x.x 0
3 × 3, 0 moves for Left and 0 for Right		 P The second dimension is not the deep end
xox/o.x/.xo ∗
3 × 3, 1 moves for Left and 1 for Right		 N The second dimension is not the deep end
xox/oxo/..x 0
after Black lifts rank 3 file 1		 P The two moves that are not captures
xox/oxo/x.. 0
after Black lifts rank 3 file 3		 P The two moves that are not captures
xox/oxo/x.x −1
3 × 3, 0 moves for Left and 1 for Right		 R The second dimension is not the deep end
xoxo. 2
1 moves for Left, 0 for Right		 L A game older than the theory, The two moves that are not captures
xoxo/o..x/xoxo 0
after Black lifts rank 2 file 2		 P The two moves that are not captures
xoxo/o.xo/.x.o 2
3 × 4, 1 moves for Left and 1 for Right		 L The second dimension is not the deep end
xoxo/ox../xoxo −1
after Black lifts rank 2 file 4		 R The two moves that are not captures
xoxo/ox.x/xoxo 0
3 × 4, 0 moves for Left and 1 for Right		 P The second dimension is not the deep end
xoxo/oxox/..xo 0
after Black lifts rank 3 file 1		 P The two moves that are not captures
xoxo/oxox/xo.. 0
after Black lifts rank 3 file 3		 P The two moves that are not captures
xoxo/oxox/xoxo 0
3 × 4, 0 moves for Left and 0 for Right		 P The second dimension is not the deep end
xoxox/o..xo/xoxox −1
after Black lifts rank 2 file 2		 R The two moves that are not captures
xoxox/ox..o/xoxox −1
after Black lifts rank 2 file 4		 R The two moves that are not captures
xoxox/ox.xo/xo.ox 0
after Black lifts rank 3 file 3		 P The two moves that are not captures
xoxox/ox.xo/xoxox 0
3 × 5, 0 moves for Left and 2 for Right		 P The two moves that are not captures
xoxox/oxoxo/..xox 0
after Black lifts rank 3 file 1		 P The two moves that are not captures
xoxox/oxoxo/xox.. 0
after Black lifts rank 3 file 5		 P The two moves that are not captures
xoxox/oxoxo/xox.x −1
3 × 5, 0 moves for Left and 2 for Right		 R The two moves that are not captures
xoxox/oxoxo/xoxox 0
3 × 5, 0 moves for Left and 0 for Right		 P The two moves that are not captures
xoxoxoxo. 4
1 moves for Left, 0 for Right		 L A game older than the theory
xxx.xo. 1/2
1 moves for Left, 1 for Right		 L A game older than the theory

Lasker's Nim

7 positions

PositionWorthOutcomeDrawn in
a heap of 0 0 P A period with a constant added, Grundy sequences, and where they stop being predictable, Splitting is a move, The formula is a limit
a heap of 11 ∗12 N Splitting is a move, The proof is sixteen cells
a heap of 12 ∗11 N Splitting is a move, The proof is sixteen cells
a heap of 3 ∗4 N A period with a constant added, Grundy sequences, and where they stop being predictable, Splitting is a move, The formula is a limit
a heap of 4 ∗3 N A period with a constant added, Grundy sequences, and where they stop being predictable, Splitting is a move, The formula is a limit
a heap of 7 ∗8 N A period with a constant added, Grundy sequences, and where they stop being predictable, One split is enough, Splitting is a move, The formula is a limit, The proof is sixteen cells
a heap of 8 ∗7 N Splitting is a move, The proof is sixteen cells

Maundy Cake

144 positions

PositionWorthOutcomeDrawn in
1×1 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
1×10 6 L Maundy Cake, The size of a cake
1×11 1 L Maundy Cake, The size of a cake
1×12 10 L Maundy Cake, The size of a cake
1×2 1 L Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
1×3 1 L Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
1×4 3 L Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
1×5 1 L Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
1×6 4 L Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
1×7 1 L Maundy Cake, The size of a cake
1×8 7 L Maundy Cake, The size of a cake
1×9 4 L Maundy Cake, The size of a cake
10×1 −6 R Maundy Cake, The size of a cake
10×10 0 P Maundy Cake, The size of a cake
10×11 −1 R Maundy Cake, The size of a cake
10×12 1 L Maundy Cake, The size of a cake
10×2 −1 R Maundy Cake, The size of a cake
10×3 −1 R Maundy Cake, The size of a cake
10×4 0 P Maundy Cake, The size of a cake
10×5 −1 R Maundy Cake, The size of a cake
10×6 0 P Maundy Cake, The size of a cake
10×7 −1 R Maundy Cake, The size of a cake
10×8 1 L Maundy Cake, The size of a cake
10×9 0 P Maundy Cake, The size of a cake
11×1 −1 R Maundy Cake, The size of a cake
11×10 1 L Maundy Cake, The size of a cake
11×11 0 P Maundy Cake, The size of a cake
11×12 4 L Maundy Cake, The size of a cake
11×2 0 P Maundy Cake, The size of a cake
11×3 0 P Maundy Cake, The size of a cake
11×4 1 L Maundy Cake, The size of a cake
11×5 0 P Maundy Cake, The size of a cake
11×6 1 L Maundy Cake, The size of a cake
11×7 0 P Maundy Cake, The size of a cake
11×8 3 L Maundy Cake, The size of a cake
11×9 1 L Maundy Cake, The size of a cake
12×1 −10 R Maundy Cake, The size of a cake
12×10 −1 R Maundy Cake, The size of a cake
12×11 −4 R Maundy Cake, The size of a cake
12×12 0 P Maundy Cake, The size of a cake
12×2 −4 R Maundy Cake, The size of a cake
12×3 −4 R Maundy Cake, The size of a cake
12×4 −1 R Maundy Cake, The size of a cake
12×5 −4 R Maundy Cake, The size of a cake
12×6 −1 R Maundy Cake, The size of a cake
12×7 −4 R Maundy Cake, The size of a cake
12×8 0 P Maundy Cake, The size of a cake
12×9 −1 R Maundy Cake, The size of a cake
2×1 −1 R Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
2×10 1 L Maundy Cake, The size of a cake
2×11 0 P Maundy Cake, The size of a cake
2×12 4 L Maundy Cake, The size of a cake
2×2 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
2×3 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
2×4 1 L Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
2×5 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
2×6 1 L Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
2×7 0 P Maundy Cake, The size of a cake
2×8 3 L Maundy Cake, The size of a cake
2×9 1 L Maundy Cake, The size of a cake
3×1 −1 R Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
3×10 1 L Maundy Cake, The size of a cake
3×11 0 P Maundy Cake, The size of a cake
3×12 4 L Maundy Cake, The size of a cake
3×2 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
3×3 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
3×4 1 L Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
3×5 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
3×6 1 L Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
3×7 0 P Maundy Cake, The size of a cake
3×8 3 L Maundy Cake, The size of a cake
3×9 1 L Maundy Cake, The size of a cake
4×1 −3 R Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
4×10 0 P Maundy Cake, The size of a cake
4×11 −1 R Maundy Cake, The size of a cake
4×12 1 L Maundy Cake, The size of a cake
4×2 −1 R Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
4×3 −1 R Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
4×4 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
4×5 −1 R Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
4×6 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
4×7 −1 R Maundy Cake, The size of a cake
4×8 1 L Maundy Cake, The size of a cake
4×9 0 P Maundy Cake, The size of a cake
5×1 −1 R Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
5×10 1 L Maundy Cake, The size of a cake
5×11 0 P Maundy Cake, The size of a cake
5×12 4 L Maundy Cake, The size of a cake
5×2 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
5×3 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
5×4 1 L Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
5×5 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
5×6 1 L Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
5×7 0 P Maundy Cake, The size of a cake
5×8 3 L Maundy Cake, The size of a cake
5×9 1 L Maundy Cake, The size of a cake
6×1 −4 R Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
6×10 0 P Maundy Cake, The size of a cake
6×11 −1 R Maundy Cake, The size of a cake
6×12 1 L Maundy Cake, The size of a cake
6×2 −1 R Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
6×3 −1 R Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
6×4 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
6×5 −1 R Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
6×6 0 P Cutcake, where every value is a whole number, Maundy Cake, The obvious cut is the wrong one, The size of a cake
6×7 −1 R Maundy Cake, The size of a cake
6×8 1 L Maundy Cake, The size of a cake
6×9 0 P Maundy Cake, The size of a cake
7×1 −1 R Maundy Cake, The size of a cake
7×10 1 L Maundy Cake, The size of a cake
7×11 0 P Maundy Cake, The size of a cake
7×12 4 L Maundy Cake, The size of a cake
7×2 0 P Maundy Cake, The size of a cake
7×3 0 P Maundy Cake, The size of a cake
7×4 1 L Maundy Cake, The size of a cake
7×5 0 P Maundy Cake, The size of a cake
7×6 1 L Maundy Cake, The size of a cake
7×7 0 P Maundy Cake, The size of a cake
7×8 3 L Maundy Cake, The size of a cake
7×9 1 L Maundy Cake, The size of a cake
8×1 −7 R Maundy Cake, The size of a cake
8×10 −1 R Maundy Cake, The size of a cake
8×11 −3 R Maundy Cake, The size of a cake
8×12 0 P Maundy Cake, The size of a cake
8×2 −3 R Maundy Cake, The size of a cake
8×3 −3 R Maundy Cake, The size of a cake
8×4 −1 R Maundy Cake, The size of a cake
8×5 −3 R Maundy Cake, The size of a cake
8×6 −1 R Maundy Cake, The size of a cake
8×7 −3 R Maundy Cake, The size of a cake
8×8 0 P Maundy Cake, The size of a cake
8×9 −1 R Maundy Cake, The size of a cake
9×1 −4 R Maundy Cake, The size of a cake
9×10 0 P Maundy Cake, The size of a cake
9×11 −1 R Maundy Cake, The size of a cake
9×12 1 L Maundy Cake, The size of a cake
9×2 −1 R Maundy Cake, The size of a cake
9×3 −1 R Maundy Cake, The size of a cake
9×4 0 P Maundy Cake, The size of a cake
9×5 −1 R Maundy Cake, The size of a cake
9×6 0 P Maundy Cake, The size of a cake
9×7 −1 R Maundy Cake, The size of a cake
9×8 1 L Maundy Cake, The size of a cake
9×9 0 P Maundy Cake, The size of a cake

Misère Nim

16 positions

PositionWorthOutcomeDrawn in
heaps 1 —
misère play has outcomes but no values, which is the whole difficulty		 P Misère play, Tame and wild
heaps 1, 1 —
misère play has outcomes but no values, which is the whole difficulty		 N Misère play, Tame and wild, Two misère outcomes are not enough, What a tame heap may be replaced by, What a wider pool rescues
heaps 1, 1, 1 —
misère play has outcomes but no values, which is the whole difficulty		 P A pass is not a move, Misère play, Nim, and the nim-sum, Tame and wild, The patch that generalised, The sentence that solved the other convention, Two misère outcomes are not enough, What a tame heap may be replaced by
heaps 1, 1, 1, 1 —
misère play has outcomes but no values, which is the whole difficulty		 N Misère play, Nim, and the nim-sum, The sentence that solved the other convention, What a wider pool rescues
heaps 1, 1, 1, 1, 1 —
misère play has outcomes but no values, which is the whole difficulty		 P Misère play
heaps 1, 1, 1, 1, 1, 1 —
misère play has outcomes but no values, which is the whole difficulty		 N Misère play
heaps 1, 1, 2 —
misère play has outcomes but no values, which is the whole difficulty		 N Misère play
heaps 1, 1, 2, 2 —
misère play has outcomes but no values, which is the whole difficulty		 P Misère play
heaps 1, 1, 5 —
misère play has outcomes but no values, which is the whole difficulty		 N A pass is not a move, Misère play, Nim, and the nim-sum, The patch that generalised, The sentence that solved the other convention
heaps 1, 2 —
misère play has outcomes but no values, which is the whole difficulty		 N Misère play, Two misère outcomes are not enough
heaps 1, 2, 2 —
misère play has outcomes but no values, which is the whole difficulty		 N Misère play
heaps 1, 2, 3 —
misère play has outcomes but no values, which is the whole difficulty		 P A pass is not a move, Misère play, Nim, and the nim-sum, Tame and wild, The patch that generalised, The sentence that solved the other convention, What a wider pool rescues
heaps 2 —
misère play has outcomes but no values, which is the whole difficulty		 N Misère play, Tame and wild, Two misère outcomes are not enough
heaps 2, 2 —
misère play has outcomes but no values, which is the whole difficulty		 P A pass is not a move, Misère play, Nim, and the nim-sum, Tame and wild, The patch that generalised, The sentence that solved the other convention, What a tame heap may be replaced by, What a wider pool rescues
heaps 2, 2, 2 —
misère play has outcomes but no values, which is the whole difficulty		 N Misère play
heaps 2, 3 —
misère play has outcomes but no values, which is the whole difficulty		 N What a tame heap may be replaced by

Mock Turtles

7 positions

PositionWorthOutcomeDrawn in
heads at 1, 2, 3, 4 0
a lost row, and a word of the code		 P The code names the move, The losing positions are a code
heads at 1, 2, 3, 4 of 12 0 P A row of coins is already a sum
heads at 1, 2, 3, 4 of 8 0 P The losing positions are a code
heads at 1, 2, 5, 6 0
a lost row, and a word of the code		 P The code names the move, The losing positions are a code
heads at 2, 4, 7 of 12 ∗8 N A row of coins is already a sum
heads at 2, 5, 8 of 8 ∗4 N The code names the move
no heads 0
a lost row, and a word of the code		 P The code names the move, The losing positions are a code

Mogul

4 positions

PositionWorthOutcomeDrawn in
heads at 1, 2, 3, 4 0
a lost row, and a word of the code		 P The losing positions are a code
heads at 1, 2, 5, 6 0
a lost row, and a word of the code		 P The losing positions are a code
heads at 1, 8 of 14 0 P A row of coins is already a sum
no heads 0
a lost row, and a word of the code		 P The losing positions are a code

Moore's Nim, k = 2

7 positions

PositionWorthOutcomeDrawn in
heaps 1, 1, 1 0
the column rule mod 3 decides the outcome and does not give this value		 P The patch that generalised
heaps 1, 3, 6 ∗10
the column rule mod 3 decides the outcome and does not give this value		 N The rule a smaller move breaks
heaps 2, 2, 2 0
the column rule mod 3 decides the outcome and does not give this value		 P The patch that generalised
heaps 2, 4, 6 ∗12
the column rule mod 3 decides the outcome and does not give this value		 N Taking from several heaps at once
heaps 3, 5, 6 ∗14
the column rule mod 3 decides the outcome and does not give this value		 N Taking from several heaps at once
heaps 3, 5, 6, 7 0
the column rule mod 3 decides the outcome and does not give this value		 P Taking from several heaps at once
heaps 5, 5, 5 0
the column rule mod 3 decides the outcome and does not give this value		 P Taking from several heaps at once

Moore's Nim, k = 3

1 position

PositionWorthOutcomeDrawn in
heaps 1, 1, 1, 1 0
the column rule mod 4 decides the outcome and does not give this value		 P Taking from several heaps at once, The rule a smaller move breaks

Nim

69 positions

PositionWorthOutcomeDrawn in
heap 0 0
equal to that Nim heap in every sum, which is what the table checks		 P Every impartial game is a Nim heap
heap 1 ∗1
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 2 ∗2
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 3 ∗3
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 4 ∗4
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 5 ∗5
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 6 ∗6
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 7 ∗7
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 8 ∗8
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heaps 1 ∗1 N Misère play, Tame and wild
heaps 1, 1 0
2 bits in binary against 2 counters		 P Nim is easy, in binary, Misère play, Tame and wild, Two misère outcomes are not enough, What a tame heap may be replaced by, What a value leaves out, What a wider pool rescues
heaps 1, 1, 1 ∗1 N A pass is not a move, Nim is easy, in binary, Misère play, Nim, and the nim-sum, Tame and wild, The patch that generalised, The sentence that solved the other convention, The theorem that needed none of the theory, Two misère outcomes are not enough, What a tame heap may be replaced by
heaps 1, 1, 1, 1 0 P Misère play, Nim, and the nim-sum, The sentence that solved the other convention, What a wider pool rescues
heaps 1, 1, 1, 1, 1 ∗1 N Misère play
heaps 1, 1, 1, 1, 1, 1 0 P Misère play
heaps 1, 1, 2 ∗2 N Misère play
heaps 1, 1, 2, 2 0 P Misère play
heaps 1, 1, 5 ∗5 N A pass is not a move, Misère play, Nim, and the nim-sum, The patch that generalised, The sentence that solved the other convention
heaps 1, 2 ∗3
3 bits in binary against 3 counters		 N Nim is easy, in binary, Misère play, Two misère outcomes are not enough
heaps 1, 2, 2 ∗1 N Misère play
heaps 1, 2, 3 0 P A pass is not a move, A token on a graph, The move that gives counters back, Nim is easy, in binary, Misère play, Nim, and the nim-sum, Tame and wild, The patch that generalised, The sentence that solved the other convention, The theorem that needed none of the theory, Three players and no answer, What a value leaves out, What a wider pool rescues
heaps 1, 2, 3, 1 ∗1 N A pass is not a move
heaps 1, 3, 6 ∗4 N No two heaps alike
heaps 1, 4 ∗5 N No two heaps alike
heaps 10, 20, 30 0
14 bits in binary against 60 counters		 P Nim is easy, in binary
heaps 100, 200, 300 ∗384
24 bits in binary against 600 counters		 N Nim is easy, in binary
heaps 1000, 1000, 1000 ∗1000
30 bits in binary against 3000 counters		 N Nim is easy, in binary
heaps 1000, 2000, 3000 ∗3968
33 bits in binary against 6000 counters		 N Nim is easy, in binary
heaps 1000000, 2000000, 3000000 ∗3932160
63 bits in binary against 6000000 counters		 N Nim is easy, in binary
heaps 1024, 1024, 1024 ∗1024
33 bits in binary against 3072 counters		 N Nim is easy, in binary
heaps 1048576, 1048576, 1048576 ∗1048576
63 bits in binary against 3145728 counters		 N Nim is easy, in binary
heaps 16, 16, 16 ∗16
15 bits in binary against 48 counters		 N Nim is easy, in binary
heaps 2 ∗2 N Misère play, Tame and wild, Two misère outcomes are not enough
heaps 2, 1, 2 ∗1 N Taking from the ends
heaps 2, 2 0 P A pass is not a move, Misère play, Nim, and the nim-sum, Tame and wild, The patch that generalised, The sentence that solved the other convention, What a tame heap may be replaced by, What a value leaves out, What a wider pool rescues, Who moves last
heaps 2, 2, 2 ∗2 N Misère play
heaps 2, 3 ∗1
4 bits in binary against 5 counters		 N Nim is easy, in binary, What a tame heap may be replaced by
heaps 256, 256, 256 ∗256
27 bits in binary against 768 counters		 N Nim is easy, in binary
heaps 3, 3 0
2 to 6 moves long		 P What a value leaves out
heaps 3, 3, 6, 6 0
4 to 18 moves long		 P What a value leaves out
heaps 3, 4, 5 ∗2
8 bits in binary against 12 counters		 N Nim is easy, in binary
heaps 3, 5 ∗6
5 bits in binary against 8 counters		 N Nim is easy, in binary
heaps 3, 5, 6 0 P Nim, and the nim-sum, Taking from several heaps at once, A row of coins is already a sum
heaps 3, 5, 7 ∗1 N Nim, and the nim-sum, The theorem that needed none of the theory
heaps 4, 4 0 P Nim, and the nim-sum, What a value leaves out
heaps 4, 4, 4 ∗4
9 bits in binary against 12 counters		 N Nim is easy, in binary
heaps 5 ∗5
3 bits in binary against 5 counters		 N Nim is easy, in binary
heaps 5, 5 0
6 bits in binary against 10 counters		 P Nim is easy, in binary, What a value leaves out
heaps 5, 5, 5 ∗5
9 bits in binary against 15 counters		 N Nim is easy, in binary
heaps 5, 5, 5, 5 0
12 bits in binary against 20 counters		 P Nim is easy, in binary
heaps 5, 5, 5, 5, 5 ∗5
15 bits in binary against 25 counters		 N Nim is easy, in binary
heaps 5, 6, 3 0
3 to 14 moves long		 P What a value leaves out
heaps 6 ∗6 N Nim, and the nim-sum
heaps 6, 6 0
2 to 12 moves long		 P What a value leaves out
heaps 64, 64, 64 ∗64
21 bits in binary against 192 counters		 N Nim is easy, in binary
heaps 7, 11, 13 ∗1
11 bits in binary against 31 counters		 N Nim is easy, in binary
heaps 7, 7 0
2 to 14 moves long		 P What a value leaves out
heaps 8, 8 0
2 to 16 moves long		 P What a value leaves out
heaps 8, 9, 1 0
3 to 18 moves long		 P What a value leaves out
Nim heap of 11 ∗11
12 positions walked for a value with 12 nodes		 N The heap is not the position
Nim, a heap of 1 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild, The genus of a sum, The rule the symbols follow
Nim, a heap of 2 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, The genus of a sum, The rule the symbols follow
Nim, a heap of 3 genus 3^31
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, The genus of a sum, The rule the symbols follow
Nim, a heap of 4 genus 4^46
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, The genus of a sum, The rule the symbols follow
Nim, a heap of 5 genus 5^57
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, The genus of a sum, The rule the symbols follow
Nim, a heap of 6 genus 6^64
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, The genus of a sum, The rule the symbols follow
Nim, a heap of 7 genus 7^75
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, The genus of a sum, The rule the symbols follow
Nim, a heap of 8 genus 8^8·10
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N The genus of a sum
Nim, a heap of 9 genus 9^9·11
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N The genus of a sum

NoGo

23 positions

PositionWorthOutcomeDrawn in
.../.b./... {1 | ∗} L Every group must keep breathing
...bb.... {2 | 0, {{1∗ | ∗} | 0, {0 | −2}}}
the ends do not add		 N How thick a wall has to be
...br.... 1 | −1
the ends add		 N How thick a wall has to be
..b.. 0 P A wall that bends, Every group must keep breathing, When the regions add
..r.. 0 P A wall that bends, Every group must keep breathing, When the regions add
.b... {2 | ∗} L A wall that bends, Every group must keep breathing, When the regions add
.b../.b.. 0 P When the regions add
.br.. ∗ N When the regions add
.r... {∗ | −2} R A wall that bends, Every group must keep breathing, When the regions add
1×1 0 P Every group must keep breathing
1×2 ∗ N Every group must keep breathing
1×3 1 | −1 N Every group must keep breathing
1×4 0 P Every group must keep breathing
1×5 ∗ N Every group must keep breathing
1×6 1 | −1 N Every group must keep breathing
1×7 {1∗ | −1∗} N Every group must keep breathing
1×8 ∗ N Every group must keep breathing
2×2 ∗ N Every group must keep breathing
2×3 0 P Every group must keep breathing
2×4 0 P Every group must keep breathing
3×3 ∗ N Every group must keep breathing
b.b.. 1 | 0 N Every group must keep breathing
br... 1 | −1 N When the regions add

Node Kayles

1 position

PositionWorthOutcomeDrawn in
a path of three ∗2 N One rule makes it cold, the other hot

Number

59 positions

PositionWorthOutcomeDrawn in
−1 −1
born on day 1		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−1/16 −1/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
−1/2 −1/2
born on day 2		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−1/4 −1/4
born on day 3		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−1/8 −1/8
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−11/16 −11/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
−11/4 −11/4
born on day 5		 P The recursion this site cannot run, The simplest game above both
−11/8 −11/8
born on day 5		 P The recursion this site cannot run, The simplest game above both
−13/16 −13/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
−13/8 −13/8
born on day 5		 P The recursion this site cannot run, The simplest game above both
−15/16 −15/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
−15/8 −15/8
born on day 5		 P The recursion this site cannot run, The simplest game above both
−2 −2
born on day 2		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−3 −3
born on day 3		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−3/16 −3/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
−3/2 −3/2
born on day 3		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−3/4 −3/4
born on day 3		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−3/8 −3/8
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−4 −4
born on day 4		 P The day a number is born
−5/16 −5/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
−5/2 −5/2
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−5/4 −5/4
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−5/8 −5/8
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−7/16 −7/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
−7/4 −7/4
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−7/8 −7/8
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
−9/16 −9/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
−9/4 −9/4
born on day 5		 P The recursion this site cannot run, The simplest game above both
−9/8 −9/8
born on day 5		 P The recursion this site cannot run, The simplest game above both
0 0
born on day 0		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
1 1
born on day 1		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
1/16 1/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
1/2 1/2
born on day 2		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
1/4 1/4
born on day 3		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
1/8 1/8
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
11/16 11/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
11/4 11/4
born on day 5		 P The recursion this site cannot run, The simplest game above both
11/8 11/8
born on day 5		 P The recursion this site cannot run, The simplest game above both
13/16 13/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
13/8 13/8
born on day 5		 P The recursion this site cannot run, The simplest game above both
15/16 15/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
15/8 15/8
born on day 5		 P The recursion this site cannot run, The simplest game above both
2 2
born on day 2		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
3 3
born on day 3		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
3/16 3/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
3/2 3/2
born on day 3		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
3/4 3/4
born on day 3		 P The birthday of a sum, The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
3/8 3/8
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
4 4
born on day 4		 P The day a number is born
5/16 5/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
5/2 5/2
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
5/4 5/4
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
5/8 5/8
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
7/16 7/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
7/4 7/4
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
7/8 7/8
born on day 4		 P The day a number is born, The numbers came out of the game, The recursion this site cannot run, The simplest game above both
9/16 9/16
born on day 5		 P The recursion this site cannot run, The simplest game above both
9/4 9/4
born on day 5		 P The recursion this site cannot run, The simplest game above both
9/8 9/8
born on day 5		 P The recursion this site cannot run, The simplest game above both

Octal game ·137

5 positions

PositionWorthOutcomeDrawn in
heap 52 ∗3
the first values of the certificate block, which repeat at heap 86		 N A chess problem that turned out to be an octal game, What the arithmetic cost in 1956
heap 53 ∗3
the first values of the certificate block, which repeat at heap 87		 N A chess problem that turned out to be an octal game, What the arithmetic cost in 1956
heap 54 0
the first values of the certificate block, which repeat at heap 88		 P A chess problem that turned out to be an octal game, What the arithmetic cost in 1956
heap 55 ∗1
the first values of the certificate block, which repeat at heap 89		 N A chess problem that turned out to be an octal game, What the arithmetic cost in 1956
heap 56 ∗1
the first values of the certificate block, which repeat at heap 90		 N A chess problem that turned out to be an octal game, What the arithmetic cost in 1956

Octal game ·77

5 positions

PositionWorthOutcomeDrawn in
heap 71 ∗7
the first values of the certificate block, which repeat at heap 83		 N Four values, and the sequence is settled for ever
heap 72 ∗4
the first values of the certificate block, which repeat at heap 84		 N Four values, and the sequence is settled for ever
heap 73 ∗1
the first values of the certificate block, which repeat at heap 85		 N Four values, and the sequence is settled for ever
heap 74 ∗2
the first values of the certificate block, which repeat at heap 86		 N Four values, and the sequence is settled for ever
heap 75 ∗8
the first values of the certificate block, which repeat at heap 87		 N Four values, and the sequence is settled for ever

Partizan

14 positions

PositionWorthOutcomeDrawn in
2 × 3 2 | −1/2
equal to no nimber up to ∗8		 N Where the impartial theory stops
2 × 4 {{2 | 0} | 0}
equal to no nimber up to ∗8		 R Where the impartial theory stops
3 × 3, 4 squares taken 1∗
equal to no nimber up to ∗8		 L Where the impartial theory stops
a switch: Left to 1, Right to −1 1 | −1
equal to no nimber up to ∗6		 N Confused is not the same as unknown, Where the impartial theory stops
a switch: Left to 2, Right to −2 2 | −2
equal to no nimber up to ∗8		 N Where the impartial theory stops
Left to 0, Right to 1 1/2
equal to no nimber up to ∗6		 L Confused is not the same as unknown, Where the impartial theory stops
Left to 2, Right to 0 2 | 0
equal to no number on the scale drawn		 N Where the impartial theory stops
star ∗
equal to no number on the scale drawn		 N Where the impartial theory stops
star two ∗2
equal to exactly one nimber in the range drawn		 N Where the impartial theory stops
that switch plus a star {1∗ | −1∗}
equal to no nimber up to ∗8		 N Where the impartial theory stops
the asymmetry at the root ↑∗
equal to no nimber up to ∗8		 N Where the impartial theory stops
the asymmetry one level down {0, ↑∗ | 0, ∗}
equal to no nimber up to ∗8		 N Where the impartial theory stops
the asymmetry two levels down {0, {0, ↑∗ | 0, ∗}, ∗ | 0, ∗, ∗2}
equal to no nimber up to ∗8		 N Where the impartial theory stops
up ↑
equal to no nimber up to ∗6		 L Confused is not the same as unknown, Where the impartial theory stops

Partizan subtraction {1,2,3} vs {1,2,3}

5 positions

PositionWorthOutcomeDrawn in
heap of 0 0 P Two players, two lists
heap of 1 ∗ N Two players, two lists
heap of 2 ∗2 N Two players, two lists
heap of 3 ∗3 N Two players, two lists
heap of 4 0 P Two players, two lists

Partizan subtraction {1,2,3} vs {2,4}

5 positions

PositionWorthOutcomeDrawn in
heap of 0 0 P A sequence with a rule and no period
heap of 1 1 L A sequence with a rule and no period
heap of 2 1 | 0 N A sequence with a rule and no period
heap of 3 1∗ L A sequence with a rule and no period
heap of 4 {1, 1∗ | 0, {1 | 0}} N A sequence with a rule and no period

Partizan subtraction {1,2} vs {1,3}

7 positions

PositionWorthOutcomeDrawn in
heap of 0 0 P A sequence with a rule and no period, Two players, two lists
heap of 1 ∗ N A sequence with a rule and no period, Two players, two lists
heap of 2 ↑ L A sequence with a rule and no period, Two players, two lists
heap of 3 ↑∗ N A sequence with a rule and no period, Two players, two lists
heap of 4 {↑ | ∗} L A sequence with a rule and no period, The condition that survived the wider sweep, Two players, two lists
heap of 6 ⇑ L A sequence with a rule and no period, Two players, two lists
heap of 9 3·↑∗ L A sequence with a rule and no period, Two players, two lists

Partizan subtraction {1,2} vs {2,3}

5 positions

PositionWorthOutcomeDrawn in
heap of 0 0 P The condition that survived the wider sweep
heap of 1 1 L The condition that survived the wider sweep
heap of 2 1 | 0 N The condition that survived the wider sweep
heap of 3 {1, {1 | 0} | 0} N The condition that survived the wider sweep
heap of 4 0 P The condition that survived the wider sweep

Partizan subtraction {1,3} vs {2}

5 positions

PositionWorthOutcomeDrawn in
heap of 0 0 P A sequence with a rule and no period
heap of 1 1 L A sequence with a rule and no period
heap of 2 1 | 0 N A sequence with a rule and no period
heap of 3 1/2 L A sequence with a rule and no period
heap of 4 {1 | {1 | 0}} L A sequence with a rule and no period, The condition that survived the wider sweep, Two players, two lists

Partizan subtraction {1} vs {2}

5 positions

PositionWorthOutcomeDrawn in
heap of 0 0 P Two players, two lists
heap of 1 1 L Two players, two lists
heap of 2 1 | 0 N Two players, two lists
heap of 3 0 P Two players, two lists
heap of 4 1 L The condition that survived the wider sweep, Two players, two lists

Partizan subtraction {2,3} vs {1,4}

5 positions

PositionWorthOutcomeDrawn in
heap of 0 0 P Two players, two lists
heap of 1 −1 R Two players, two lists
heap of 2 0 | −1 N Two players, two lists
heap of 3 {0 | {0 | −1}} L Two players, two lists
heap of 4 −1/2 R The condition that survived the wider sweep, Two players, two lists

Partizan subtraction {3} vs {3,5}

5 positions

PositionWorthOutcomeDrawn in
heap of 0 0 P The condition that survived the wider sweep
heap of 1 0 P The condition that survived the wider sweep
heap of 2 0 P The condition that survived the wider sweep
heap of 3 ∗ N The condition that survived the wider sweep
heap of 4 ∗ N The condition that survived the wider sweep

Poker Nim

4 positions

PositionWorthOutcomeDrawn in
heaps 1, 2, 3, reserves 6/1 0
the reserves are drawn, and they are not part of the value		 P The move that gives counters back
heaps 1, 2, 4, reserves 2/2 ∗7
the reserves are drawn, and they are not part of the value		 N The move that gives counters back
heaps 3, 4, 5, reserves 2/2 ∗2
the reserves are drawn, and they are not part of the value		 N What a component has to carry
heaps 3, 5, 7, reserves 4/4 ∗1
the reserves are drawn, and they are not part of the value		 N The move that gives counters back, Taking from several heaps at once, The condition the recursion rests on

Push

20 positions

PositionWorthOutcomeDrawn in
...LLLR −73/512 R The criterion that cannot exist
...LLR −21/64 R The other way to move a row
...RRL 21/64 L The other way to move a row
..LL.R −3/8
a strip the count gets wrong		 R The reading that survives too much
..LLLR −9/64 R The criterion that cannot exist, The other way to move a row
..RR.L 3/8
a strip the count gets wrong		 L The reading that survives too much
..RRRL 9/64 L The other way to move a row
.L 1 L Nothing worth fighting over, The other way to move a row, The reading that survives too much
.L.L.R −1/2
a strip the count gets wrong		 R The reading that survives too much
.LL.R −1/2
a strip the count gets wrong		 R The reading that survives too much
.R −1 R The other way to move a row
.RR.L 1/2
a strip the count gets wrong		 L The reading that survives too much
L 0 P Nothing worth fighting over, The other way to move a row, The reading that survives too much
L.R −1 R Nothing worth fighting over, The other way to move a row, The reading that survives too much
LL 0 P The other way to move a row
LR 0 P Nothing worth fighting over, The other way to move a row, The reading that survives too much
R 0 P The other way to move a row
R.LL 2 L The other way to move a row, The reading that survives too much
RL.L 1 L The other way to move a row, The reading that survives too much
RLL 0 P The other way to move a row, The reading that survives too much

Random turns

10 positions

PositionWorthOutcomeDrawn in
{0, {1 | 0} | {∗ | −1}} — Left's chance 1/2 N The best chance is the wrong move
{0, {1 | 0} | ↓} — Left's chance 9/16 N The best chance is the wrong move
↑ — Left's chance 1/2 L Left always wins, and loses more often than not, The coldest position has the biggest swing
∗ — Left's chance 1/2 N A coin needs no tie-break
0 — Left's chance 1/2 P A coin needs no tie-break, The best chance is the wrong move, The coldest position has the biggest swing
1 — Left's chance 3/4 L A coin needs no tie-break, The coldest position has the biggest swing
1 | −1 — Left's chance 1/2 N A coin needs no tie-break, The coldest position has the biggest swing
1 | 0 — Left's chance 5/8 N The best chance is the wrong move
2 — Left's chance 7/8 L Left always wins, and loses more often than not
3 — Left's chance 15/16 L Left always wins, and loses more often than not

Ruler

4 positions

PositionWorthOutcomeDrawn in
heads at 1, 3 0
a lost row, and a word of the code		 P The losing positions are a code
heads at 1, 5 0
a lost row, and a word of the code		 P The losing positions are a code
heads at 6, 9, 12 of 14 ∗7 N A row of coins is already a sum
no heads 0
a lost row, and a word of the code		 P The losing positions are a code

Shove

24 positions

PositionWorthOutcomeDrawn in
..L 3 L Below zero, Nothing worth fighting over, The other way to move a row
..LRL 15/4 L Nobody comes back, What the class does not buy
..R −3 R Below zero, Nothing worth fighting over, The other way to move a row
.LL 5 L Below zero, Nothing worth fighting over, The other way to move a row
.LR −2 R Below zero, Nothing worth fighting over, The other way to move a row
.RL 2 L Nothing worth fighting over
L 1 L Nothing worth fighting over
L.L 4 L Nothing worth fighting over, The reading that survives too much
L.R −5/2 R Nothing worth fighting over, The reading that survives too much
LL 3 L Nothing worth fighting over, The reading that survives too much
LLL 6 L Nothing worth fighting over, The reading that survives too much
LLLL 10 L Nothing worth fighting over
LLLR −15/8 R Nothing worth fighting over
LLR −7/4 R Nothing worth fighting over, The reading that survives too much
LLRR −23/4 R Nobody comes back, Nothing worth fighting over, What the class does not buy
LR −3/2 R Nothing worth fighting over, The reading that survives too much
LRL 9/4 L Nothing worth fighting over
LRLR −23/8 R Nobody comes back, Nothing worth fighting over, What the class does not buy
LRR −9/2 R Nothing worth fighting over, The reading that survives too much
LRRR −17/2 R Nothing worth fighting over
RL 3/2 L Nothing worth fighting over
RRL 7/4 L Nothing worth fighting over
RRRL 15/8 L Nothing worth fighting over
RRRRL 31/16 L Nothing worth fighting over

Snort

11 positions

PositionWorthOutcomeDrawn in
a bowtie 4 | −4 N One rule makes it cold, the other hot
a bowtie from L.... 3 | 0 N One board, two rules
a path of four {{2 | 1} | {−1 | −2}} N One board, two rules
a path of four from .L.. 2 | 1 L One rule makes it cold, the other hot
a path of three 2 | −2 N One board, two rules, One rule makes it cold, the other hot
a path of three from .L. 2 L One board, two rules
a path of three from L.. 1 | 0 N One board, two rules
a star 3 | −3 N One board, two rules
a star from ..LR ∗ N One rule makes it cold, the other hot
a star from L... 3 L One rule makes it cold, the other hot
a triangle 2 | −2 N One board, two rules

Subtraction 1,2

5 positions

PositionWorthOutcomeDrawn in
heap 1 ∗1 N Four values, and the sequence is settled for ever
heap 2 ∗2 N Four values, and the sequence is settled for ever
heap 3 0 P Four values, and the sequence is settled for ever
heap 4 ∗1 N Four values, and the sequence is settled for ever
heap 5 ∗2 N Four values, and the sequence is settled for ever

Subtraction 1,2,3

5 positions

PositionWorthOutcomeDrawn in
heap 1 ∗1 N Four values, and the sequence is settled for ever
heap 2 ∗2 N Four values, and the sequence is settled for ever
heap 3 ∗3 N Four values, and the sequence is settled for ever
heap 4 0 P Four values, and the sequence is settled for ever
heap 5 ∗1 N Four values, and the sequence is settled for ever

Subtraction 1,3,4

5 positions

PositionWorthOutcomeDrawn in
heap 1 ∗1 N Four values, and the sequence is settled for ever, Two players, two lists
heap 2 0 P Four values, and the sequence is settled for ever, Two players, two lists
heap 3 ∗1 N Four values, and the sequence is settled for ever, Two players, two lists
heap 4 ∗2 N Four values, and the sequence is settled for ever, Two players, two lists
heap 5 ∗3 N Four values, and the sequence is settled for ever, Two players, two lists

Subtraction 2,5,6

5 positions

PositionWorthOutcomeDrawn in
heap 1 0 P Four values, and the sequence is settled for ever, Take one, three or four
heap 2 ∗1 N Four values, and the sequence is settled for ever, Take one, three or four
heap 3 ∗1 N Four values, and the sequence is settled for ever, Take one, three or four
heap 4 0 P Four values, and the sequence is settled for ever, Take one, three or four
heap 5 ∗2 N Four values, and the sequence is settled for ever, Take one, three or four

Subtraction 2,5,7

5 positions

PositionWorthOutcomeDrawn in
heap 1 0 P The period is small and the proof does not say so
heap 2 ∗1 N The period is small and the proof does not say so
heap 3 ∗1 N The period is small and the proof does not say so
heap 4 0 P The period is small and the proof does not say so
heap 5 ∗2 N The period is small and the proof does not say so

Switch

34 positions

PositionWorthOutcomeDrawn in
{−1 | −2} −1 | −2
mean −3/2, temperature 1/2		 R A fight with no midpoint
{−1 | −3} −1 | −3
mean −2, temperature 1		 R When a switch is not a switch
{−1/2 | −1} −1/2 | −1
mean −3/4, temperature 1/4		 R A fight with no midpoint
{0 | −1} 0 | −1
mean −1/2, temperature 1/2		 N When a switch is not a switch
{0 | −1/2} 0 | −1/2
mean −1/4, temperature 1/4		 N A fight with no midpoint
{0 | −2} 0 | −2
mean −1, temperature 1		 N When a switch is not a switch
{1 | −1} 1 | −1
mean 0, temperature 1		 N How hot a day gets, Worth nothing, and worth fighting for, The thirty that cancel themselves, What has to break before a pawn is worth a number, When a switch is not a switch
{1 | 0} 1 | 0
mean 1/2, temperature 1/2		 N A fight with no midpoint, A number and a fight, One board, two rules, How hot a day gets, Worth nothing, and worth fighting for, The thirty that cancel themselves
{1 | 1/2} 1 | 1/2
mean 3/4, temperature 1/4		 L A fight with no midpoint
{1/2 | −1/2} 1/2 | −1/2
mean 0, temperature 1/2		 N Cooling adds and heating does not, How hot a background has to be, Worth nothing, and worth fighting for
{1/2 | 0} 1/2 | 0
mean 1/4, temperature 1/4		 N A fight with no midpoint
{1/32 | −1/32} 1/32 | −1/32
mean 0, temperature 1/32		 N Worth nothing, and worth fighting for
{1/8 | −1/8} 1/8 | −1/8
mean 0, temperature 1/8		 N How hot a background has to be, Worth nothing, and worth fighting for
{2 | −1} 2 | −1
mean 1/2, temperature 3/2		 N When a switch is not a switch
{2 | −1/2} 2 | −1/2
mean 3/4, temperature 5/4		 N When a switch is not a switch
{2 | −2} 2 | −2
mean 0, temperature 2		 N A fight with no midpoint, One board, two rules, How hot a background has to be, How hot a day gets, Worth nothing, and worth fighting for, The thirty that cancel themselves, What has to break before a pawn is worth a number, When a switch is not a switch
{2 | −3/2} 2 | −3/2
mean 1/4, temperature 7/4		 N When a switch is not a switch
{2 | 0} 2 | 0
mean 1, temperature 1		 N A number and a fight, Cooling adds and heating does not, When a switch is not a switch
{2 | 1} 2 | 1
mean 3/2, temperature 1/2		 L A fight with no midpoint, When a switch is not a switch
{2 | 1/2} 2 | 1/2
mean 5/4, temperature 3/4		 L When a switch is not a switch
{2 | 3/2} 2 | 3/2
mean 7/4, temperature 1/4		 L When a switch is not a switch
{3 | −1} 3 | −1
mean 1, temperature 2		 N A number and a fight, Cooling adds and heating does not, Worth nothing, and worth fighting for, When a switch is not a switch
{3 | −3} 3 | −3
mean 0, temperature 3		 N One board, two rules, What has to break before a pawn is worth a number
{3 | 1} 3 | 1
mean 2, temperature 1		 L When a switch is not a switch
{3/2 | 1/2} 3/2 | 1/2
mean 1, temperature 1/2		 L Cooling adds and heating does not
{4 | −4} 4 | −4
mean 0, temperature 4		 N One board, two rules
{4 | 0} 4 | 0
mean 2, temperature 2		 N A fight with no midpoint, A number and a fight, Worth nothing, and worth fighting for
{4 | 2} 4 | 2
mean 3, temperature 1		 L Worth nothing, and worth fighting for
{5 | 1} 5 | 1
mean 3, temperature 2		 L Worth nothing, and worth fighting for
{5 | 4} 5 | 4
mean 9/2, temperature 1/2		 L A number and a fight
{6 | 0} 6 | 0
mean 3, temperature 3		 N Worth nothing, and worth fighting for
{7 | −1} 7 | −1
mean 3, temperature 4		 N Worth nothing, and worth fighting for
{7/2 | 5/2} 7/2 | 5/2
mean 3, temperature 1/2		 L Worth nothing, and worth fighting for
{8 | −8} 8 | −8
mean 0, temperature 8		 N Worth nothing, and worth fighting for

Sylver Coinage

18 positions

PositionWorthOutcomeDrawn in
⟨2, 3⟩ 0
1 gaps, Frobenius number 1		 P A parity with a first exception, The game that is a number system
⟨2, 5⟩ ∗1
2 gaps, Frobenius number 3		 N Two ways to end with no bound
⟨3, 4⟩ ∗2
3 gaps, Frobenius number 5		 N Two ways to end with no bound
⟨3, 5⟩ ∗3
4 gaps, Frobenius number 7		 N The game that is a number system
⟨3, 7, 11⟩ ∗3
5 gaps, Frobenius number 8		 N Every move closes the largest gap
⟨3, 7⟩ ∗4
6 gaps, Frobenius number 11		 N Every move closes the largest gap, The pairing removes moves it cannot name
⟨4, 5, 11⟩ 0
5 gaps, Frobenius number 7		 P The pairing removes moves it cannot name
⟨4, 5, 6, 7⟩ 0
3 gaps, Frobenius number 3		 P A parity with a first exception
⟨4, 5, 7⟩ ∗2
4 gaps, Frobenius number 6		 N Every move closes the largest gap
⟨4, 5⟩ ∗3
6 gaps, Frobenius number 11		 N The game that is a number system
⟨4, 7⟩ ∗5
9 gaps, Frobenius number 17		 N The game that is a number system, Two ways to end with no bound
⟨4, 9, 14, 15⟩ 0
8 gaps, Frobenius number 11		 P A parity with a first exception
⟨4, 9⟩ ∗8
12 gaps, Frobenius number 23		 N Two ways to end with no bound
⟨5, 6⟩ ∗7
10 gaps, Frobenius number 19		 N Two ways to end with no bound
⟨5, 7, 9, 11⟩ ∗5
7 gaps, Frobenius number 13		 N A shortlist with nothing at the top, The pairing removes moves it cannot name
⟨5, 7⟩ ∗8
12 gaps, Frobenius number 23		 N The game that is a number system, Two ways to end with no bound
⟨5, 9⟩ ∗10
16 gaps, Frobenius number 31		 N The game that is a number system
⟨7, 11⟩ ∗20
30 gaps, Frobenius number 59		 N The game that is a number system

Toads

3 positions

PositionWorthOutcomeDrawn in
Toads and Frogs TT..FF ∗
70 positions walked for a value with 2 nodes		 N The heap is not the position, The strip nobody has a formula for
Toads and Frogs TT.TFF {1 | ∗}
16 positions walked for a value with 4 nodes		 L The strip nobody has a formula for
Toads and Frogs TTT...FFF 0
1129 positions walked for a value with 1 node		 P Knowing who wins, and knowing what it is worth, The strip nobody has a formula for, A position reached eleven ways is one position

Toads and Frogs

29 positions

PositionWorthOutcomeDrawn in
T. 1 L Toads and Frogs
T..F 0 P The class where nobody runs out first, Toads and Frogs
T..FF −1 R The strip nobody has a formula for
T.F ∗ N The strip nobody has a formula for, Toads and Frogs
T.F. 1/2 L The class where nobody runs out first, The same strip without the jump
T.FF 0 | −1/2 N The strip nobody has a formula for, Toads and Frogs
T.FFF {0 | −1∗} N The strip nobody has a formula for
T.FT.F 0 P The strip nobody has a formula for, Toads and Frogs
T.TF 1/2 L The same strip without the jump, The strip nobody has a formula for, Toads and Frogs
T.TFF ↑ L Toads and Frogs
TF. 0 P Toads and Frogs
TF.. 1 L The class where nobody runs out first
TF.TF 0 P The strip nobody has a formula for
TT. 2 L Toads and Frogs
TT...F 2∗ L The strip nobody has a formula for
TT..F 1 L The strip nobody has a formula for
TT..FF ∗ N Toads and Frogs
TT.F 1/2 | 0 N The same strip without the jump, The strip nobody has a formula for, Toads and Frogs
TT.FF ∗ N The strip nobody has a formula for
TT.TFF {1 | ∗} L The strip nobody has a formula for
TTF. 0 P The strip nobody has a formula for
TTF..F ∗ N The class where nobody runs out first, Toads and Frogs
TTF.F ↓ R Toads and Frogs
TTF.FT ↓ R The strip nobody has a formula for
TTT. 3 L Toads and Frogs
TTT... 9 L Toads and Frogs
TTT...F 4∗ L The strip nobody has a formula for
TTT...FFF 0 P The same strip without the jump, The strip nobody has a formula for
TTT..FFF 1/8 | −1/8 N The strip nobody has a formula for

Top Entails

13 positions

PositionWorthOutcomeDrawn in
a Top Entails heap of 1 no nimber N A move that must be answered, A pass is not a move, What a component has to carry
a Top Entails heap of 2 no nimber
loony: a win for the mover whatever else is on the board		 N A move that must be answered, A pass is not a move, What a component has to carry
a Top Entails heap of 3 no nimber P A move that must be answered, A pass is not a move, What a component has to carry
a Top Entails heap of 4 no nimber
loony: a win for the mover whatever else is on the board		 N A move that must be answered, A pass is not a move, What a component has to carry
a Top Entails heap of 6 no nimber
loony: a win for the mover whatever else is on the board		 N A move that must be answered
a Top Entails heap of 8 no nimber
loony: a win for the mover whatever else is on the board		 N A move that must be answered
Top Entails heaps of 2 and 2 no nimber
the nim-sum predicts P		 N A move that must be answered
Top Entails heaps of 2 and 4 no nimber
the nim-sum predicts P		 N A move that must be answered
Top Entails heaps of 2 and 6 no nimber
the nim-sum predicts P		 N A move that must be answered
Top Entails heaps of 4 and 2 no nimber
the nim-sum predicts P		 N A move that must be answered
two Top Entails heaps of 2 no nimber
a position added to itself that the first player wins		 N A move that must be answered
two Top Entails heaps of 4 no nimber
a position added to itself that the first player wins		 N A move that must be answered
two Top Entails heaps of 6 no nimber
a position added to itself that the first player wins		 N A move that must be answered

Toppling Dominoes

26 positions

PositionWorthOutcomeDrawn in
empty 0 P A recipe instead of a census, How long a row a value needs, Topple it from either end
G ∗ N How long a row a value needs
GG ∗2 N How long a row a value needs
GGG ∗3 N How long a row a value needs
GRG {0 | 0, {0 | −1}} N How long a row a value needs
L 1 L How long a row a value needs, Topple it from either end
LGL 1∗ L How long a row a value needs
LGR 1 | −1 N How long a row a value needs
LGRG {1 | 0, {0 | −1}} N How long a row a value needs
LLLLLLL 7 L Topple it from either end
LLLLLLLR 6 | 0 N Topple it from either end
LLR 1 | 0 N How long a row a value needs, The other way to move a row
LLRLRLL 1↑ L How long a row a value needs
LLRR 1 | −1 N The other way to move a row, Topple it from either end
LR ∗ N The other way to move a row, Topple it from either end
LRL 1/2 L How long a row a value needs, Topple it from either end
LRR 0 | −1 N The other way to move a row
LRRL ↑ L A recipe instead of a census, Topple it from either end
LRRLL {1 | ∗} L Topple it from either end
LRRR 0 | −2 N Topple it from either end
R −1 R How long a row a value needs
RG 0 | −1 N How long a row a value needs
RLRR −1∗ R Topple it from either end
RRL 0 | −1 N A recipe instead of a census, Topple it from either end
RRLR −1∗ R Topple it from either end
RRRL 0 | −2 N Topple it from either end

Turning Turtles

6 positions

PositionWorthOutcomeDrawn in
heads at 1, 2, 3 0
a lost row, and a word of the code		 P The losing positions are a code
heads at 1, 4, 5 0
a lost row, and a word of the code		 P The losing positions are a code
heads at 3, 5, 6 of 12 0 P A row of coins is already a sum
heads at 3, 5, 8 of 12 ∗14 N The tartan theorem, A row of coins is already a sum
heads at 8 of 12 ∗8 N A row of coins is already a sum
no heads 0
a lost row, and a word of the code		 P The losing positions are a code

Value

360 positions

PositionWorthOutcomeDrawn in
-1 −1 R How rare it is to be bigger, Misère play has no negatives, Numbers avoid numbers, The first theorem, and the winner it declines to name, The values that are their own negatives, Three players and no answer, Who moves last, Start at the end and work backwards
-1 + (−-1) 0 P Misère play has no negatives
-3/2 −3/2 R Nothing worth fighting over
{ | ↓} −1 R An option nobody would take
{ | −1} −2
0 with −1 added		 R An option nobody would take
{ | ∗} 0 P The reduction that puts options back
{ | 0} −1
0 with 0 added		 R An option nobody would take
{-1|-2} −1 | −2 R Who moves last
{-2|-1} −3/2 R When a switch is not a switch
{{∗ | −1} | −2} {{∗ | −1} | −2}
one option a side		 R A fight with no midpoint, The bend is the condition
{{0 | ∗2} | 0} ↑∗ N Canonical form, The reduction that puts options back
{{0, ∗ | −1} | −2} {{0, ∗ | −1} | −2}
one option a side		 R A fight with no midpoint, The bend is the condition
{{1 | −1}, ∗2 | −2} {{1 | −1}, ∗2 | −2}
mean −1, temperature 1, residue ↑∗		 R A number and a fight
{{1 | ∗}, ↑∗ | {∗ | −1}, ↓∗} {{1 | ∗}, ↑∗ | {∗ | −1}, ↓∗}
equal to its own negative		 N A self-negative value costs a day, The thirty that cancel themselves, The values that are their own negatives, Where the order and the sum disagree
{{1 | 0, ∗} | −1/2} {{1 | 0, ∗} | −1/2}
mean 0, temperature 1/2, residue ↓		 R A number and a fight
{{1 | 0}, {1 | ∗} | {0 | −1}, {∗ | −1}} {{1 | 0}, {1 | ∗} | {0 | −1}, {∗ | −1}}
equal to its own negative		 N A self-negative value costs a day, The thirty that cancel themselves, The values that are their own negatives, Where the order and the sum disagree
{{1 | 1} | {−1 | −1}} {1∗ | −1∗} N The values that are their own negatives
{{1 | 1} | {−1 | −1}} + (−{{1 | 1} | {−1 | −1}}) 0 P The values that are their own negatives
{{1|1} | 0} {1∗ | 0} N Cooling adds and heating does not
{{2 | 0} | 0} {{2 | 0} | 0} R Turn the board through a right angle
{{2 | 0} | 0} + (−{{2 | 0} | 0}) 0 P Turn the board through a right angle
{{2 | 1} | {0 | −1}} {{2 | 1} | {0 | −1}} L Nobody wants to move here
{{2|0} | 0} {{2 | 0} | 0}
cools by 1 to 0		 R The operator chosen for one game
{↑ | −1/2} {↑ | −1/2} N What a number does to a fight
{↑ | ∗} {↑ | ∗} L When the ups add
{↑ | ∗} + ↓ {0 | ↓∗} L When the ups add
{↑ | 1} 1/2 L The reduction that puts options back
{↑∗, ↑ | ↓∗, ↓} {↑, ↑∗ | ↓, ↓∗} N The values that are their own negatives
{↑∗, ↑ | ↓∗, ↓} + (−{↑∗, ↑ | ↓∗, ↓}) 0 P The values that are their own negatives
{↓ | ↑} 0 P Equal in every company
{⇑ | ↓} {⇑ | ↓} N When the ups add
{⇑ | ↓} + {⇑ | ↓} ↑ L When the ups add
{−1 | −1/2} −3/4
the number −3/4 — no temperature		 R A fight with no midpoint
{−1 | 0} −1/2
the number −1/2 — no temperature		 R A fight with no midpoint
{−1 | 1} 0 P Cutcake, where every value is a whole number, Equal in every company
{−1, 0, ∗ | 1} 1/2 L Equal in every company
{−1, 0, 1 | 1, 2} 1∗ L The move that gives counters back, Two hundred and fifty-six ways to write twenty-two things
{−1/2 | 0} −1/4
the number −1/4 — no temperature		 R A fight with no midpoint
{−1/2, {1 | −1} | −1} {−1/2, {1 | −1} | −1}
mean −3/4, temperature 1/4, residue ∗		 R A number and a fight
{−1/8 | 1/8} 0 P The simplicity rule
{−2 | −1} −3/2
the number −3/2 — no temperature		 R A fight with no midpoint
{−2 | 2} 0 P Maundy Cake, When a switch is not a switch
{−5 | 5} 0 P The simplicity rule
{−5 | 7} 0 P The simplicity rule
{−6 | 2} 0 P The simplicity rule
{∗ | ↓} {∗ | ↓} R The class where nobody runs out first, When the ups add
{∗ | ↓} + {∗ | ↓} {{0 | 0, ↓∗} | 0} R When the ups add
{∗ | ↓} + ↑ {↑∗ | 0} R The class where nobody runs out first, When the ups add
{∗ | ↓} + ∗ {0 | 0, ↓∗} N When the ups add
{∗ | ↓} + Clobber 1×3 xxo {↑∗ | 0} R When the ups add
{∗ | ↓} + green EE {∗3 | {∗3 | 0}} R When the ups add
{∗ | −1} {∗ | −1}
stops 0 and −1		 R The fight never runs backwards
{∗ | ∗} 0 P Equal in every company, How old a value is
{∗, ↑ | 0, ↑} ↑∗ N Canonical form
{∗2 | −1/2} {∗2 | −1/2} R What a number does to a fight
{∗2 | −2} {∗2 | −2} R What a number does to a fight
{∗2 | ∗2} 0 P Equal in every company
{0 | -2} 0 | −2 N Which part to move in
{0 | {0 | −1}} {0 | {0 | −1}} L Infinitesimals
{0 | {0 | −2}} {0 | {0 | −2}} L Equal in every company, Infinitesimals
{0 | {0 | −2}} + {{2 | 0} | 0} 0 P Equal in every company
{0 | {0 | −4}} {0 | {0 | −4}} L Equal in every company
{0 | {0 | −4}} + {{2 | 0} | 0} {{2 | 0}, {{2 | 0} | 0} | {{{2 | 0} | 0} | {{−2 | −4} | −4}}} R Equal in every company
{0 | ↓} ∗ N The reduction that puts options back
{0 | 0} ∗
cools by 1 to 0		 N Cooling adds and heating does not, When a switch is not a switch
{0 | 1} 1/2
the number 1/2 — no temperature		 L A fight with no midpoint, Equal in every company, The simplicity rule, When a switch is not a switch
{0 | 1/2} 1/4
the number 1/4 — no temperature		 L A fight with no midpoint, The simplicity rule
{0 | 1/4} 1/8 P The simplicity rule
{0 | 1/8} 1/16 P The simplicity rule
{0 | 10} 1 P The simplicity rule
{0 | 2} 1 P Cutcake, where every value is a whole number, When a switch is not a switch
{0 | 3} 1 P The simplicity rule
{0 | 6} 1 P Maundy Cake
{0, -1 | 1, 2} 1/2 L What a value leaves out
{0, -1 | 1} 1/2 L A reduction that reads a graph, Canonical form, Comparing two positions means playing a third, The notation was the argument, The reduction that puts options back, What a value costs to write down
{0, -3 | 1} 1/2 L Canonical form
{0, {0 | −1} | 0, {1 | 0}} ∗ N The same position, written once, Topple it from either end
{0, {1|0} | 1} 1/2 L Canonical form
{0, ↑ | 0, ∗2, ↓} ∗ N Two hundred and fifty-six ways to write twenty-two things, The reduction that puts options back
{0, ↑∗ | 0, ↓∗} {0, ↑∗ | 0, ↓∗} N The values that are their own negatives
{0, ↑∗ | 0, ↓∗} + (−{0, ↑∗ | 0, ↓∗}) 0 P The values that are their own negatives
{0, ∗ | −1} {0, ∗ | −1}
stops 0 and −1		 N The fight never runs backwards
{0, ∗ | ∗} ↑ L The reduction that always shrinks
{0, ∗ | 0, ∗2, ↓} {0, ∗ | ↓} N The reduction that always shrinks
{1 | -1} 1 | −1 N The thirty that cancel themselves, Where the impartial theory stops
{1 | -1} + (−{1 | -1}) 0 P The thirty that cancel themselves, Where the impartial theory stops
{1 | −1} 1 | −1 N An option nobody would take, How hot a day gets, Nobody wants to move here, The operator chosen for one game, The operator that puts the star back, The values that are their own negatives
{1 | −1} + (−{1 | −1}) 0 P The values that are their own negatives
{1 | ∗} {1 | ∗} L Cooling adds and heating does not, The birthday of a sum, The fight never runs backwards
{1 | ∗} + (−{1 | ∗}) 0 P The birthday of a sum
{1 | 0, ∗} {1 | 0, ∗}
stops 1 and 0		 N The fight never runs backwards
{1 | 0} 1 | 0
cools by 1 to 1/2		 N A number and a fight, Below zero, Cooling by exactly one, Equal in every company, How hot a day gets, Numbers avoid numbers, The operator that puts the star back, What is left when the small change is thrown away
{1 | 0} + (−{1 | 0}) 0 P Equal in every company
{1 | 0} + −2 −1 | −2 R Equal in every company
{1 | 0} + 1/2 + ∗ + ↑ + 0 {3/2↑∗ | 1/2↑∗} L What is left when the small change is thrown away
{1 | 1} 1∗
cools by 1 to 1		 L Cooling adds and heating does not, When a switch is not a switch
{1 | 2} 3/2
the number 3/2 — no temperature		 L A fight with no midpoint
{1 | 3} 2 P Cutcake, where every value is a whole number
{1 | 4} 2 P The simplicity rule
{1 | 5} 2 P The simplicity rule
{1, {1 | −1} | −1, {1 | −1}} {1, {1 | −1} | −1, {1 | −1}}
equal to its own negative		 N A self-negative value costs a day, The thirty that cancel themselves
{1, {1 | −1} | −1} {1, {1 | −1} | −1}
1 | −1 with 1 | −1 added		 N An option nobody would take, Nobody wants to move here
{1, {1 | ∗} | −1, {∗ | −1}} {1, {1 | ∗} | −1, {∗ | −1}}
equal to its own negative		 N A self-negative value costs a day, The thirty that cancel themselves
{1, {1 | 0, ∗} | −1, {0, ∗ | −1}} {1, {1 | 0, ∗} | −1, {0, ∗ | −1}}
equal to its own negative		 N A self-negative value costs a day, The thirty that cancel themselves
{1, {2 | 0} | −1} {1, {2 | 0} | −1}
1 | −1 with 2 | 0 added		 N An option nobody would take
{1, 2 | −1} 2 | −1
1 | −1 with 2 added		 N An option nobody would take
{1/2 | −1/2} 1/2 | −1/2 N The values that are their own negatives
{1/2 | −1/2} + (−{1/2 | −1/2}) 0 P The values that are their own negatives
{1/2 | 1} 3/4
the number 3/4 — no temperature		 L A fight with no midpoint
{1/2 | 1/2} 1/2∗
worth 1/2∗ — temperature 0, and not a number		 L When a switch is not a switch
{1/2, {1 | 0, ∗} | −1/2, {0, ∗ | −1}} {1/2, {1 | 0, ∗} | −1/2, {0, ∗ | −1}}
mean 0, temperature 1/2, residue ∗2		 N A number and a fight
{1/4 | 1} 1/2 P The simplicity rule
{1/4 | 1/2} 3/8 P The simplicity rule
{1/4 | 3/4} 1/2 P The simplicity rule
{1/4 | 3/8} 5/16 P The simplicity rule
{1|-1} 1 | −1 N Misère play has no negatives, Who moves last
{1|-1} + (−{1|-1}) 0 P Misère play has no negatives
{1|0} 1 | 0 N Two misère outcomes are not enough
{1|0} + (−{1|0}) 0 P Two misère outcomes are not enough
{1∗ | −1∗} {1∗ | −1∗}
equal to its own negative		 N A self-negative value costs a day, The thirty that cancel themselves, The values that are their own negatives, Where the order and the sum disagree
{10 | {9 | 1}} {10 | {9 | 1}}
cools by 1 to {9 | {9 | 3}}		 L A number and a fight, Cooling by exactly one, The operator that puts the star back
{16 | 0} 16 | 0
mean 8, temperature 8, residue ∗		 N A number and a fight
{2 | {1 | −1}, ↓} {2 | {1 | −1}, ↓}
mean 1, temperature 1, residue ↓∗		 N A number and a fight
{2 | {1 | −1}, ∗2} {2 | {1 | −1}, ∗2}
mean 1, temperature 1, residue ↓∗		 L A number and a fight
{2 | {1 | −1}} {2 | {1 | −1}}
mean 1, temperature 1, residue ↑		 L A number and a fight, What a number does to a fight
{2 | {1 | 0}} {2 | {1 | 0}} L What a number does to a fight
{2 | {3|1}} 3 L Canonical form
{2 | −1/2} 2 | −1/2
cools by 1 to 1 | 1/2		 N The operator chosen for one game
{2 | −2} 2 | −2
cools by 1 to 1 | −1		 N How hot a day gets, The values that are their own negatives
{2 | −2} + (−{2 | −2}) 0 P The values that are their own negatives
{2 | 0} 2 | 0
cools by 1 to 1∗		 N A number and a fight, Below zero, Cooling adds and heating does not, Cooling by exactly one, Equal in every company, Turn the board through a right angle, Misère play has no negatives, Numbers avoid numbers, One part that never ends, The company that is closed, The endgame, accounted for, The operator that puts the star back, The values that are their own negatives, What can be struck out
{2 | 0} + -1 1 | −1 N Numbers avoid numbers
{2 | 0} + (−{2 | 0}) 0 P Turn the board through a right angle, Misère play has no negatives, One part that never ends, The company that is closed, The values that are their own negatives, What can be struck out
{2 | 0} + {1 | 0} {{3 | 2} | {1 | 0}} L Numbers avoid numbers
{2 | 0} + −2 0 | −2 N Equal in every company
{2 | 0} + 1 3 | 1 L Numbers avoid numbers
{2 | 11/4} 5/2 P When a switch is not a switch
{2 | 13/4} 3
the number 3 — no temperature		 L When a switch is not a switch
{2 | 15/4} 3
the number 3 — no temperature		 L When a switch is not a switch
{2 | 17/4} 3
the number 3 — no temperature		 L When a switch is not a switch
{2 | 19/4} 3
the number 3 — no temperature		 L When a switch is not a switch
{2 | 2} 2∗
worth 2∗ — temperature 0, and not a number		 L When a switch is not a switch
{2 | 3} 5/2
the number 5/2 — no temperature		 L When a switch is not a switch
{2 | 4} 3 P Cutcake, where every value is a whole number
{2 | 5} 3 P When a switch is not a switch
{2 | 5/2} 9/4
the number 9/4 — no temperature		 L When a switch is not a switch
{2 | 7/2} 3
the number 3 — no temperature		 L When a switch is not a switch
{2 | 9/2} 3
the number 3 — no temperature		 L When a switch is not a switch
{2, 0 | 1} 2 | 1 L A reduction that reads a graph
{2, 0 | 5, {3|1}} 3 L Canonical form
{2|-2} 2 | −2 N When a switch is not a switch
{2|0} 2 | 0 N What a wider pool rescues
{2|0} + (−{2|0}) 0 P What a wider pool rescues
{2|1} 2 | 1 L When a switch is not a switch, Who moves last
{3 | −1} 3 | −1
mean 1, temperature 2, residue ∗		 N A number and a fight, Cooling by exactly one
{3 | 1} 3 | 1 L The endgame, accounted for
{3 | 20} 4 P Maundy Cake
{3 | 4} 7/2 P The simplicity rule
{3/2 | 7/4} 13/8 P The simplicity rule
{4 | {2 | 0}} {4 | {2 | 0}} L Nobody wants to move here
{4 | 0} 4 | 0
cools by 1 to 3 | 1		 N A number and a fight, Cooling by exactly one, The endgame, accounted for
{4 | 0} + {3 | 1} + {2 | 0} 7 | 3 L The endgame, accounted for
{5 | {4 | 0}} {5 | {4 | 0}}
the answer settles the fight		 L What the halving is a function of
{5/8 | 7/8} 3/4 P The simplicity rule
{6 | {4 | {3 | 1}}} {6 | {4 | {3 | 1}}}
the answer starts another fight		 L What the halving is a function of
{6 | 0} 6 | 0
cools by 2 to 4 | 2		 N Cooling by exactly one
{7/8 | 9/8} 1 P The simplicity rule
* ∗ N Two people, four years apart, one theorem
*2 ∗2 N The values that are their own negatives
↑ ↑
cools by 1 to 0		 L A number and a fight, The class where nobody runs out first, Below zero, Cooling adds and heating does not, Cooling by exactly one, Equal in every company, Turn the board through a right angle, Infinitesimals, Misère play has no negatives, Nobody has to move, Nobody wants to move here, One part that never ends, Three ways to add the same games, Outcomes do not add, The birthday of a sum, The company that is closed, The fight never runs backwards, The simplest game above both, The sum is the object, Toads and Frogs, What a number does to a fight, What a wider pool rescues, What an infinitesimal does to a fight, What can be struck out, What is left when the small change is thrown away, When the ups add, Which part to move in, Who moves last
↑ + (−↑) 0 P Turn the board through a right angle, Misère play has no negatives, One part that never ends, The birthday of a sum, The company that is closed, What a wider pool rescues, What can be struck out
↑ + ⇑ 3·↑ L When the ups add
↑ + ⇓ ↓ R The sum is the object
↑ + ∗ ↑∗ N Equal in every company, Three ways to add the same games, When the ups add, Which part to move in
↑ + 1 + {0 | -2} {1↑ | −1↑} N Which part to move in
↑ − −1/64 1/64↑ L Confused is not the same as unknown
↑ − ∗ ↑∗ N Comparing positions, Confused is not the same as unknown, When the ups add
↑ − ∗2 {0 | ∗3} L When the ups add
↑ − 0 ↑ L How many ups, Comparing positions, Confused is not the same as unknown, Turn the board through a right angle
↑ − 1/1024 −1/1024↑ R Comparing positions
↑ − 1/64 −1/64↑ R Confused is not the same as unknown
↑ ∨ ∗, inside day three ⇑
the same question, larger company		 L How rare it is to be bigger, The simplest game above both
↑ ∨ ∗, inside day two 1/2
a least upper bound		 L How rare it is to be bigger, The simplest game above both
↑↑ ⇑ L Infinitesimals
↑↑↑ 3·↑ L Equal in every company, Infinitesimals
↑↑↑ + ∗ 3·↑∗ L Equal in every company
↑↑↑↑ 4·↑ L Infinitesimals
↑↑↑∗ 3·↑∗ L When the ups add
↑↑↑∗ + ↑∗ 4·↑ L When the ups add
↑∗ ↑∗ N Infinitesimals, Nobody has to move, Nobody wants to move here, Outcomes do not add, The fight never runs backwards, Toads and Frogs, Two misère outcomes are not enough, What a number does to a fight, What an infinitesimal does to a fight, When the ups add
↑∗ + (−↑∗) 0 P Two misère outcomes are not enough
↑∗ + ↑∗ ⇑ L When the ups add
↑∗ + ↓∗ 0 P When the ups add
↑∗ − ↑ ∗ N Comparing positions
↑∗ − ∗ ↑ L How many ups, Comparing positions, Turn the board through a right angle
↑∗ − 0 ↑∗ N Confused is not the same as unknown, When the ups add
↓ ↓
cools by 1 to 0		 R Cooling adds and heating does not, Infinitesimals, Nobody has to move, Nobody wants to move here, Outcomes do not add, The fight never runs backwards, Toads and Frogs, What an infinitesimal does to a fight, Who moves last
↓↓ ⇓ R Infinitesimals
↓↓↓ 3·↓ R Infinitesimals
↓↓↓↓ 4·↓ R Infinitesimals
↓∗ ↓∗
stops 0 and 0		 N The fight never runs backwards
⇑ ⇑ L Infinitesimals, Outcomes do not add, Toads and Frogs, What a number does to a fight, What an infinitesimal does to a fight, When the ups add
⇑ + ↓∗ ↑∗ N When the ups add
⇑ − ↑ ↑ L How many ups, Comparing positions, Confused is not the same as unknown, Turn the board through a right angle
⇑ − ∗ 2·↑∗ L Comparing positions, When the ups add
⇑ − 1/1024 −1/1024⇑ R Comparing positions
⇑ − 1/64 −1/64⇑ R Confused is not the same as unknown
⇑∗ 2·↑∗ L When the ups add
⇑∗ + ⇑∗ 4·↑ L When the ups add
⇓ ⇓ R The sum is the object, When the ups add
⇓ + ⇓∗ 4·↓∗ R When the ups add
−1 −1 R A rule with no promise at all, The fight never runs backwards, The other way to move a row
−1 − 0 −1 R Comparing positions
−1 − 1 −2 R Confused is not the same as unknown
−1 − 1 | −1 0 | −2 N Confused is not the same as unknown
−1/2 −1/2
stops −1/2 and −1/2		 R The fight never runs backwards
−1/2 | −1 −1/2 | −1
one option a side		 R A fight with no midpoint, The bend is the condition
−1/2 | −2 −1/2 | −2
one option a side		 R A fight with no midpoint, The bend is the condition
−1∗ −1∗
stops −1 and −1		 R The fight never runs backwards
−2 −2
stops −2 and −2		 R The fight never runs backwards
−3 −3 R Turn the board through a right angle
−3 + (−−3) 0 P Turn the board through a right angle
−3/2 −3/2
cools by 1 to −3/2		 R The operator chosen for one game
∗ ∗ N A move that must be answered, A number and a fight, A rule with no promise at all, A self-negative value costs a day, At least five hundred and seventy-one, Below zero, Cooling adds and heating does not, Cooling by exactly one, Equal in every company, Equal in this company, Turn the board through a right angle, Fifty-two errors and seven sizes, How hot a day gets, How rare it is to be bigger, Infinitesimals, Misère play has no negatives, Nobody has to move, Nobody wants to move here, Nothing worth fighting over, Numbers avoid numbers, One part that never ends, Three ways to add the same games, Outcomes do not add, The birthday of a sum, The company that is closed, The fight never runs backwards, The first theorem, and the winner it declines to name, The other way to move a row, The simplest game above both, The thirty that cancel themselves, The values that are their own negatives, Three players and no answer, Toads and Frogs, What a number does to a fight, What a wider pool rescues, What an infinitesimal does to a fight, What can be struck out, What is left when the small change is thrown away, When the ups add, Where the impartial theory stops, Where the order and the sum disagree, Which part to move in, Who moves last, Start at the end and work backwards
∗ + (−∗) 0 P Turn the board through a right angle, Misère play has no negatives, One part that never ends, The birthday of a sum, The company that is closed, The thirty that cancel themselves, What a wider pool rescues, What can be struck out, Where the impartial theory stops
∗ + {1 | ∗} + {{1|1} | 0} 1∗ L Cooling adds and heating does not
∗ + ∗ 0 P Equal in every company, Equal in this company, When the ups add
∗ + ∗2 ∗3 N Which part to move in
∗ + ∗2 + ∗3 0 P A move that must be answered, Which part to move in
∗ + 1 1∗ L Numbers avoid numbers
∗ − ↑ ↓∗ N Comparing positions
∗ − ⇓ 2·↑∗ L When the ups add
∗ − −1/64 1/64∗ L Confused is not the same as unknown
∗ − 0 ∗ N How many ups, Comparing positions, Confused is not the same as unknown, Turn the board through a right angle, How much a list of options can lose, The same fight, eight times over, The numbers it is confused with
∗ − 1/64 −1/64∗ R Confused is not the same as unknown
∗ ∧ ↑ ↓∗
the greatest lower bound inside day two		 N The simplest game above both
∗ ∧ ∗2 ↓∗
the greatest lower bound inside day two		 N The simplest game above both
∗ ∧ 1 | −1 0 | −1
the greatest lower bound inside day two		 N Where the order and the sum disagree
∗ ∨ ↑ 1/2
the least upper bound inside day two		 L The simplest game above both
∗ ∨ ∗2 ↑∗
the least upper bound inside day two		 N The simplest game above both
∗ ∨ ∗2, inside day three {0, ∗ | 0, ↑∗}
the same question, larger company		 N Where the order and the sum disagree
∗ ∨ ∗2, inside day two ↑∗
a least upper bound		 N Where the order and the sum disagree
∗ ∨ 1 | −1 1 | 0
the least upper bound inside day two		 N Where the order and the sum disagree
∗ ∨ 1 | −1, inside day three {1 | ↓}
the same question, larger company		 N Where the order and the sum disagree
∗ ∨ 1 | −1, inside day two 1 | 0
a least upper bound		 N Where the order and the sum disagree
∗ against {1 | 0, ∗} {{1↓∗ | ↓∗}, ∗ | {↓ | −1↓}}
an error the temperature rule does not predict		 N Fifty-two errors and seven sizes
∗2 ∗2 N A move that must be answered, At least five hundred and seventy-one, Equal in every company, Equal in this company, Turn the board through a right angle, Misère play has no negatives, The fight never runs backwards, The simplest game above both, The sum is the object, The thirty that cancel themselves, The values that are their own negatives, Two misère outcomes are not enough, When the ups add, Where the impartial theory stops, Which part to move in
∗2 + (−∗2) 0 P Equal in every company, Turn the board through a right angle, Misère play has no negatives, The thirty that cancel themselves, The values that are their own negatives, Two misère outcomes are not enough, Where the impartial theory stops
∗2 + ↓ {∗3 | 0} R When the ups add
∗2 + ∗ ∗3 N Equal in every company, Equal in this company
∗2 + ∗2 0 P Equal in every company
∗2 + ∗3 ∗ N The sum is the object, When the ups add
∗2 − ∗ ∗3 N Comparing positions, How much a list of options can lose
∗2 − 0 ∗2 N Comparing positions, Confused is not the same as unknown
∗3 ∗3 N A move that must be answered, Equal in every company, Turn the board through a right angle, Misère play has no negatives, The sum is the object, Which part to move in
∗3 + (−∗3) 0 P Turn the board through a right angle, Misère play has no negatives
∗3 + ∗2 ∗ N Equal in every company
∗4 ∗4 N When the ups add
∗4 + green EE ∗6 N When the ups add
0 0 P A rule with no promise at all, A self-negative value costs a day, An option nobody would take, At least five hundred and seventy-one, Equal in every company, Fifty-two errors and seven sizes, How rare it is to be bigger, Nobody has to move, Nothing worth fighting over, The fight never runs backwards, The first theorem, and the winner it declines to name, The operator that puts the star back, The other way to move a row, The simplest game above both, The thirty that cancel themselves, The values that are their own negatives, Three players and no answer, Two people, four years apart, one theorem, What is left when the small change is thrown away, When a switch is not a switch, Where the order and the sum disagree, Who moves last, Start at the end and work backwards
0 | −1 0 | −1 N One of four questions, The fight never runs backwards, Where the order and the sum disagree
0 | −1 ∧ 1 | 0 0 | −1
the greatest lower bound inside day two		 N One of four questions, Where the order and the sum disagree
0 | −1 ∨ 1 | 0 1 | 0
the least upper bound inside day two		 N One of four questions, Where the order and the sum disagree
0 − ↑ ↓ R Comparing positions
0 − ∗ ∗ N Comparing positions
0 − 0 0 P Comparing positions
0 ∧ ↓∗ −1/2
a greatest lower bound inside day two		 R Fifty-two errors and seven sizes, How rare it is to be bigger, Where the order and the sum disagree
0 ∧ ∗ −1/2
a greatest lower bound inside day two		 R Fifty-two errors and seven sizes, How rare it is to be bigger, The simplest game above both, Where the order and the sum disagree
0 ∧ ∗2 ↓
the greatest lower bound inside day two		 R The simplest game above both
0 ∧ 0 | −1 −1∗
a greatest lower bound inside day two		 R Fifty-two errors and seven sizes, How rare it is to be bigger, Where the order and the sum disagree
0 ∧ 1 | −1 {∗ | −1}
a greatest lower bound inside day two		 R Fifty-two errors and seven sizes, How rare it is to be bigger, Where the order and the sum disagree
0 ∧ 1 | 0 ↓
a greatest lower bound inside day two		 R Fifty-two errors and seven sizes, How rare it is to be bigger, Where the order and the sum disagree
0 ∨ ↓∗ ↑
a least upper bound inside day two		 L Fifty-two errors and seven sizes, How rare it is to be bigger, Where the order and the sum disagree
0 ∨ ∗ 1/2
a least upper bound inside day two		 L Fifty-two errors and seven sizes, How rare it is to be bigger, The simplest game above both, Where the order and the sum disagree
0 ∨ ∗, inside day three {0 | {0, ∗ | −1}}
the same question, larger company		 L How rare it is to be bigger, The simplest game above both
0 ∨ ∗, inside day two 1/2
a least upper bound		 L How rare it is to be bigger, The simplest game above both
0 ∨ ∗2 ↑
the least upper bound inside day two		 L The simplest game above both
0 ∨ ∗2, inside day three {0 | ↑, ∗}
the same question, larger company		 L How rare it is to be bigger, Where the order and the sum disagree
0 ∨ ∗2, inside day two ↑
a least upper bound		 L How rare it is to be bigger, Where the order and the sum disagree
0 ∨ 0 | −1 ↑
a least upper bound inside day two		 L Fifty-two errors and seven sizes, How rare it is to be bigger, Where the order and the sum disagree
0 ∨ 1 | −1 {1 | ∗}
a least upper bound inside day two		 L Fifty-two errors and seven sizes, How rare it is to be bigger, Where the order and the sum disagree
0 ∨ 1 | 0 1∗
a least upper bound inside day two		 L Fifty-two errors and seven sizes, How rare it is to be bigger, Where the order and the sum disagree
0 against {0, ∗ | −1} {{1↑∗ | ↑} | {↑∗ | −1↑}, ∗}
an error the temperature rule does not predict		 N Fifty-two errors and seven sizes
0 against {1 | 0, ∗} {{1↓ | ↓∗}, ∗ | {↓ | −1↓∗}}
an error the temperature rule does not predict		 N Fifty-two errors and seven sizes
1 1 L A rule with no promise at all, Cooling adds and heating does not, How rare it is to be bigger, Misère play has no negatives, Nobody wants to move here, Numbers avoid numbers, One part that never ends, The fight never runs backwards, The first theorem, and the winner it declines to name, The other way to move a row, The sum is the object, The values that are their own negatives, Three players and no answer, Which part to move in, Who moves last, Start at the end and work backwards
1 + (−1) 0 P Misère play has no negatives, One part that never ends
1 + 1/2 3/2 L Numbers avoid numbers
1 | −1 1 | −1
equal to its own negative		 N A self-negative value costs a day, At least five hundred and seventy-one, The fight never runs backwards, The thirty that cancel themselves, The values that are their own negatives, What a number does to a fight, Where the order and the sum disagree
1 | −1 − ∗ {1∗ | −1∗} N How rare it is to be bigger
1 | −1 − ∗2 {1∗2 | −1∗2} N How much a list of options can lose
1 | −1 − 0 1 | −1 N Confused is not the same as unknown, How much a list of options can lose, How rare it is to be bigger
1 | −1 − 1 0 | −2 N Confused is not the same as unknown
1 | −1 − 1/2 1/2 | −3/2 N Confused is not the same as unknown
1 | 0 1 | 0 N One of four questions, The fight never runs backwards, Where the order and the sum disagree
1 | 0 ∨ 0 | −1, inside day three 1 | 0
the same question, larger company		 N The simplest game above both
1 | 0 ∨ 0 | −1, inside day two 1 | 0
a least upper bound		 N The simplest game above both
1 − 0 1 L Comparing positions
1 ∨ ∗, inside day three 1
the same question, larger company		 L The simplest game above both
1 ∨ ∗, inside day two 1
a least upper bound		 L The simplest game above both
1/2 1/2
cools by 1 to 1/2		 L Below zero, Canonical form, Comparing positions, Cooling adds and heating does not, Cooling by exactly one, Equal in every company, Turn the board through a right angle, How hot a day gets, Misère play has no negatives, Nobody comes back, Nobody wants to move here, Numbers avoid numbers, The fight never runs backwards, The operator chosen for one game, The operator that puts the star back, What is left when the small change is thrown away
1/2 + (−1/2) 0 P Equal in every company, Turn the board through a right angle, Misère play has no negatives
1/2 + ∗ 1/2∗ L Equal in every company
1/2 | −1/2 1/2 | −1/2 N Nobody wants to move here, What a number does to a fight
1/2 − 1/4 1/4 L How many ups, Comparing positions, Turn the board through a right angle, The same fight, eight times over
1/2 ∨ ∗2, inside day three 1/2
the same question, larger company		 L Where the order and the sum disagree
1/2 ∨ ∗2, inside day two 1/2
a least upper bound		 L Where the order and the sum disagree
1∗ 1∗
stops 1 and 1		 L The fight never runs backwards
1∗ − 1 ∗ N Comparing positions
2 2 L Turn the board through a right angle, Misère play has no negatives, Nobody wants to move here, The fight never runs backwards, Two misère outcomes are not enough
2 + (−2) 0 P Turn the board through a right angle, Misère play has no negatives, Two misère outcomes are not enough
2 | −1/2 2 | −1/2
one option a side		 N A fight with no midpoint, The bend is the condition
2 | −2 2 | −2
equal to its own negative		 N A self-negative value costs a day, The thirty that cancel themselves, The values that are their own negatives, What a number does to a fight, Where the order and the sum disagree
2 | −2 − 0 2 | −2 N Confused is not the same as unknown
2 | 0 2 | 0 N Nobody wants to move here, What a number does to a fight, What an infinitesimal does to a fight
2 | 0 − 0 2 | 0 N Confused is not the same as unknown, The same fight, eight times over
2 | 0 − 1 1 | −1 N Confused is not the same as unknown, The same fight, eight times over, The numbers it is confused with
2 | 0 − 2 0 | −2 N The same fight, eight times over, The numbers it is confused with
2 | 1/2 2 | 1/2
one option a side		 L A fight with no midpoint, The bend is the condition
2 − ∗ 2∗ L How rare it is to be bigger
2 − 1 1 L How rare it is to be bigger
3 3 L Two people, four years apart, one theorem
3 + * + 0 3∗ L Two people, four years apart, one theorem
3 | 0 3 | 0 N Nobody wants to move here
3 | 0 − −1 4 | 1 L Confused is not the same as unknown
3 | 0 − 3/2 3/2 | −3/2 N Confused is not the same as unknown
3 | 0 − 4 −1 | −4 R Confused is not the same as unknown
3 | 1 3 | 1 L Nobody wants to move here
3·↑ 3·↑ L Nobody wants to move here
3/2 3/2 L Nothing worth fighting over
3/4 3/4 L Turn the board through a right angle, Misère play has no negatives, When the nested sum only sees the value
3/4 + (−3/4) 0 P Turn the board through a right angle, Misère play has no negatives
3/8 3/8 L Nobody wants to move here
4 | −4 4 | −4 N What a number does to a fight
4 | 2 4 | 2 L Nobody wants to move here
6 | 0 6 | 0 N Nobody wants to move here
clobber 1×3 xxo ↑ L The sum is the object
Clobber 1×3 xxo + Clobber 1×2 xo ↑∗ N When the ups add
clobber 1×3 xxo + green E + 1 1↑∗ L The sum is the object
green E ∗ N The sum is the object, Which part to move in
green EE ∗2 N Which part to move in
green EE + green EE 0 P When the ups add
green LR 1/2 L Which part to move in
green LR + green RL + green E + green EE ∗3 N Which part to move in
green RL −1/2 R Which part to move in
miny-1 {{1 | 0} | 0} R A pawn ending is a sum, Tiny, miny, and the sizes below every size
miny-2 {{2 | 0} | 0} R A pawn ending is a sum, Tiny, miny, and the sizes below every size
miny-4 {{4 | 0} | 0} R A pawn ending is a sum, Tiny, miny, and the sizes below every size
Shove .L.LLL 17
born on day 17 and shown in 6 pieces		 L The birthday is a floor, The cheapest way to show a value
Shove .LRRRR −17
born on day 17 and shown in 6 pieces		 R The birthday is a floor, The cheapest way to show a value
Shove LR.LLL 57/4
born on day 17 and shown in 6 pieces		 L The birthday is a floor, The cheapest way to show a value
the discrepancy at 0, ↓∗ {−1/2 | −1/2↑}
what the join and the meet leave over		 R Fifty-two errors and seven sizes
the discrepancy at 0, ∗ ∗
what the join and the meet leave over		 N Fifty-two errors and seven sizes
the discrepancy at 0, 0 | −1 {↑∗ | −1↑∗}
what the join and the meet leave over		 R Fifty-two errors and seven sizes
the discrepancy at 0, 1 | 0 {1↓∗ | ↓∗}
what the join and the meet leave over		 L Fifty-two errors and seven sizes
tiny-1 {0 | {0 | −1}} L A pawn ending is a sum, Tiny, miny, and the sizes below every size
tiny-16 {0 | {0 | −16}} L Tiny, miny, and the sizes below every size
tiny-2 {0 | {0 | −2}} L A pawn ending is a sum, Tiny, miny, and the sizes below every size
tiny-3 {0 | {0 | −3}} L Tiny, miny, and the sizes below every size
tiny-4 {0 | {0 | −4}} L A pawn ending is a sum, Tiny, miny, and the sizes below every size
tiny-8 {0 | {0 | −8}} L Tiny, miny, and the sizes below every size

Welter

19 positions

PositionWorthOutcomeDrawn in
0, 1 0 P No two heaps alike
0, 1, 2 0 P No two heaps alike
0, 1, 4, 5 0 P No two heaps alike
0, 2, 4, 6 0 P No two heaps alike
0, 2, 5, 7 0 P No two heaps alike
0, 3 ∗2 N No two heaps alike
0, 3, 4 0 P No two heaps alike
0, 6 ∗5 N No two heaps alike
1, 3, 5 0 P No two heaps alike
1, 3, 6 ∗7 N No two heaps alike
1, 4 ∗4 N No two heaps alike
1, 4, 6 0 P No two heaps alike
2, 3 0 P No two heaps alike
2, 3, 8, 9 0 P No two heaps alike
2, 4, 5 0 P No two heaps alike
2, 5 ∗6 N No two heaps alike
3, 7 ∗3 N No two heaps alike
4, 5 0 P No two heaps alike
6, 7 0 P No two heaps alike

Wythoff

9 positions

PositionWorthOutcomeDrawn in
heaps 1, 2 0
numerals 1 and 10		 P The digits say which move wins, The heap is not the position, Wythoff's game, and the ratio nobody put there
heaps 3, 5 0
numerals 100 and 1000		 P The digits say which move wins, The heap is not the position, Wythoff's game, and the ratio nobody put there
heaps 4, 7 0
numerals 101 and 1010		 P The digits say which move wins, The heap is not the position, Wythoff's game, and the ratio nobody put there
heaps 6, 10 0
numerals 1001 and 10010		 P The digits say which move wins, The heap is not the position, Wythoff's game, and the ratio nobody put there
queen at (1, 2) 0
a cold position, by both routes		 P A golden ratio thirty years early, Three complete solutions in nine years
queen at (3, 5) 0
a cold position, by both routes		 P A golden ratio thirty years early, Three complete solutions in nine years
queen at (4, 7) 0
a cold position, by both routes		 P A golden ratio thirty years early, Three complete solutions in nine years
queen at (6, 10) 0
a cold position, by both routes		 P A golden ratio thirty years early, Three complete solutions in nine years, Wythoff's game, and the ratio nobody put there
queen at (8, 13) 0
a cold position, by both routes		 P A golden ratio thirty years early, Three complete solutions in nine years

Wythoff's game

1 position

PositionWorthOutcomeDrawn in
(5, 8) ∗2 N The digits say which move wins, Wythoff's game, and the ratio nobody put there

a fight with two follow-ups

4 positions

PositionWorthOutcomeDrawn in
{{12 | 2} | {1 | −3}} {{12 | 2} | {1 | −3}}
temperature 3, follow-ups 12 | 2 and 1 | −3		 L Double sente is not a property of the position
{{5 | 3} | {2 | −4}} {{5 | 3} | {2 | −4}}
temperature 2, follow-ups 5 | 3 and 2 | −4		 L Double sente is not a property of the position
{{6 | 2} | {1 | −3}} {{6 | 2} | {1 | −3}}
temperature 5/2, follow-ups 6 | 2 and 1 | −3		 L Double sente is not a property of the position
{{8 | 2} | {1 | −3}} {{8 | 2} | {1 | −3}}
temperature 3, follow-ups 8 | 2 and 1 | −3		 L Double sente is not a property of the position

a game in an environment

14 positions

PositionWorthOutcomeDrawn in
{{5 | 3} | {2 | −4}} {{5 | 3} | {2 | −4}}
mean 2, temperature 2		 L An environment made of coupons, What a move nobody makes is worth
{10 | {9 | 1}} {10 | {9 | 1}}
mean 9, temperature 1		 L An environment made of coupons
{2 | {1 | 0}} {2 | {1 | 0}}
mean 5/4, temperature 3/4		 L Two games in one environment
{2 | {1 | 0}}, beside 4 | 0 {2 | {1 | 0}}
entered at 0 in company, 1 alone		 L Two games in one environment
{4 | {3 | −3}} {4 | {3 | −3}}
mean 3, temperature 1		 L How big the answer is
{4 | {3 | 0}} {4 | {3 | 0}}
mean 3, temperature 1		 L When to leave the environment
{5 | {4 | 0}} {5 | {4 | 0}}
mean 4, temperature 1		 L An environment made of coupons
{5 | {4 | 0}}, beside 4 | 0 {5 | {4 | 0}}
entered at 2 in company, 1 alone		 L The answer that starts another fight, Two games in one environment
2 | 0, beside 4 | 0 2 | 0
entered at 2 in company, 1 alone		 N Two games in one environment, When to leave the environment
3/2 3/2
mean 3/2, temperature −1		 L An environment made of coupons
4 | 0 4 | 0
mean 2, temperature 2		 N An environment made of coupons
4 | 0, beside {2 | {1 | 0}} 4 | 0
entered at 2 in company, 2 alone		 N Two games in one environment
4 | 0, beside {5 | {4 | 0}} 4 | 0
entered at 2 in company, 2 alone		 N The answer that starts another fight, Two games in one environment
4 | 0, beside 2 | 0 4 | 0
entered at 2 in company, 2 alone		 N Two games in one environment, When to leave the environment

a local fight

11 positions

PositionWorthOutcomeDrawn in
{{4 | 2} | 0} {{4 | 2} | 0}
temperature 3/2, follow-up 0		 N What a move nobody makes is worth
{{5 | 3} | {2 | −4}} {{5 | 3} | {2 | −4}}
temperature 2, follow-up 2 | −4		 L Double sente is not a property of the position, What a move nobody makes is worth
{10 | {9 | 1}} {10 | {9 | 1}}
temperature 1, follow-up 9 | 1		 L Sente is a fact about the rest of the board
{2 | {1 | 0}} {2 | {1 | 0}}
temperature 3/4, follow-up 1 | 0		 L Sente is a fact about the rest of the board
{4 | {3 | 1}} {4 | {3 | 1}}
temperature 1, follow-up 3 | 1		 L The answer that starts another fight
{5 | {4 | 0}} {5 | {4 | 0}}
temperature 1, follow-up 4 | 0		 L A rule with no promise at all, Sente is a fact about the rest of the board, Two games in one environment, What a move nobody makes is worth
{6 | {2 | {1 | −9}}} {6 | {2 | {1 | −9}}}
temperature 5/2, follow-up {2 | {1 | −9}}		 L What the halving is a function of
{6 | {4 | {3 | 1}}} {6 | {4 | {3 | 1}}}
temperature 3/2, follow-up {4 | {3 | 1}}		 L The answer that starts another fight
{8 | {2 | −20}} {8 | {2 | −20}}
temperature 6, follow-up 2 | −20		 L A pool built to punish greed
{8 | {4 | {3 | {2 | 0}}}} {8 | {4 | {3 | {2 | 0}}}}
temperature 5/2, follow-up {4 | {3 | {2 | 0}}}		 L A subtraction, not a factor
4 | 0 4 | 0
temperature 2, follow-up 0		 N Sente is a fact about the rest of the board

difference game

54 positions

PositionWorthOutcomeDrawn in
{{2|1}|{0|−1}} − {{2|1}|{0|−1}} 0
44 positions decide the comparison		 P Comparing two positions means playing a third
{{4|2}|0} − 0 {{4 | 2} | 0}
7 positions decide the comparison		 N What a move is worth to the player making it
{{6|2}|{1|-3}} − {4|-1} 2 | −2
70 positions decide the comparison		 N The switch a player is imagining
{0,−1|1} − {0|1} 0
11 positions decide the comparison		 P Comparing two positions means playing a third
{0,−1|1} − 1/2 0
11 positions decide the comparison		 P Comparing two positions means playing a third
{1 | 0} − {1/2 | 0} {{1 | 1/2} | {0 | −1/2}}
12 positions decide the comparison		 L The same strip without the jump
{1|0} − {1|0} 0
9 positions decide the comparison		 P Comparing two positions means playing a third
{10|{9|1}} − {9|1} {8, {9 | 1} | 0}
143 positions decide the comparison		 N What a move is worth to the player making it
{2 | 0} − {1 | 0} {{2 | 1} | {0 | −1}}
12 positions decide the comparison		 L Equal in every company
{2 | 0} − 1 1 | −1
8 positions decide the comparison		 N The same strip without the jump
{2|0} − {1|0} {{2 | 1} | {0 | −1}}
12 positions decide the comparison		 L Comparing two positions means playing a third
{2|0} − {2|0} 0
16 positions decide the comparison		 P Comparing two positions means playing a third
{2|0} − 1 1 | −1
8 positions decide the comparison		 N Comparing two positions means playing a third, Knowing who wins, and knowing what it is worth, What a move is worth to the player making it
{3|-1} − {3|-1} 0
29 positions decide the comparison		 P The switch a player is imagining
{3|0} − 0 3 | 0
5 positions decide the comparison		 N What a move is worth to the player making it
{4|0} − 0 4 | 0
6 positions decide the comparison		 N What a move is worth to the player making it
{4|2} − {{4|2}|0} {2, {4 | 2} | 0}
42 positions decide the comparison		 N What a move is worth to the player making it
{5|{4|0}} − {4|0} {4, {5 | 1} | 0}
48 positions decide the comparison		 N The answer that starts another fight, What a move is worth to the player making it
{5|1} − {5|1} 0
49 positions decide the comparison		 P The switch a player is imagining
{5|1} − 1 4 | 0
14 positions decide the comparison		 N The answer that starts another fight, What a move is worth to the player making it
{6|0} − 0 6 | 0
8 positions decide the comparison		 N What a move is worth to the player making it
↑ − ↑ 0
8 positions decide the comparison		 P Comparing two positions means playing a third
↑ − ∗ ↑∗
5 positions decide the comparison		 N Comparing two positions means playing a third, The fight never runs backwards, What a move is worth to the player making it
↑ − ∗2 {0 | ∗3}
8 positions decide the comparison		 L The simplest game above both
↑ − 0 ↑
3 positions decide the comparison		 L Knowing who wins, and knowing what it is worth, The fight never runs backwards
↑ − 1/2 −1/2↑
9 positions decide the comparison		 R Equal in every company
↑∗ − ↑∗ 0
16 positions decide the comparison		 P Comparing two positions means playing a third
⇑ − ∗ 2·↑∗
9 positions decide the comparison		 L The fight never runs backwards
∗ − ∗ 0
3 positions decide the comparison		 P Comparing two positions means playing a third
∗ − ∗2 ∗3
5 positions decide the comparison		 N Equal in every company, The simplest game above both
∗ − 0 ∗
2 positions decide the comparison		 N Comparing two positions means playing a third, The fight never runs backwards, What a move is worth to the player making it
∗ − 1/2 −1/2∗
6 positions decide the comparison		 R The same strip without the jump
0 − −1 1
2 positions decide the comparison		 L Comparing two positions means playing a third
0 − ∗ ∗
2 positions decide the comparison		 N The same strip without the jump, The simplest game above both
0 − 1 −1
2 positions decide the comparison		 R What a move is worth to the player making it
0 − 1/2 −1/2
3 positions decide the comparison		 R What a move is worth to the player making it
1 − {1|0} 1 | 0
6 positions decide the comparison		 N What a move is worth to the player making it
1 − 0 1
2 positions decide the comparison		 L The same strip without the jump
1 − 2 −1
6 positions decide the comparison		 R What a move is worth to the player making it
1/2 − {0 | 1} 0
9 positions decide the comparison		 P Equal in every company
1/2 − ∗ 1/2∗
6 positions decide the comparison		 L The simplest game above both
1/2 − 0 1/2
3 positions decide the comparison		 L The simplest game above both
1/2 − 1/4 1/4
12 positions decide the comparison		 L Comparing two positions means playing a third
1/2 − 3/4 −1/4
12 positions decide the comparison		 R Comparing two positions means playing a third, What a move is worth to the player making it
1/8 − 1/16 1/16
30 positions decide the comparison		 L Comparing two positions means playing a third
10 − {10|{9|1}} {{9 | 1} | 0}
143 positions decide the comparison		 N What a move is worth to the player making it
2 − {2|0} 2 | 0
12 positions decide the comparison		 N What a move is worth to the player making it
2 − 1 1
6 positions decide the comparison		 L Comparing two positions means playing a third
3 − {3|0} 3 | 0
20 positions decide the comparison		 N What a move is worth to the player making it
3/4 − 1/2 1/4
12 positions decide the comparison		 L Comparing two positions means playing a third
4 − {4|0} 4 | 0
30 positions decide the comparison		 N What a move is worth to the player making it
5 − {5|{4|0}} {{5 | 1} | 0}
48 positions decide the comparison		 N The answer that starts another fight, What a move is worth to the player making it
5 − {5|1} 4 | 0
42 positions decide the comparison		 N The answer that starts another fight, What a move is worth to the player making it
6 − {6|0} 6 | 0
56 positions decide the comparison		 N What a move is worth to the player making it

pawn ending

12 positions

PositionWorthOutcomeDrawn in
gap 1 + gap 1 0
level material		 P A pawn ending is a sum
gap 1 + gap 2 + gap 3 0
level material		 P A pawn ending is a sum
gap 1 + gap 3, White may double ⇑
level material		 L A pawn ending is a sum
gap 1, contested + gap 1, contested 0
level material		 P One king, and two files to be in
gap 2, contested + gap 1 {1∗ | −1∗}
level material		 N A position with no value, and the rule that gives it one
gap 2, contested + gap 1, Black stuck 2 | 0
level material		 N What has to break before a pawn is worth a number
gap 2, contested + gap 3, contested + gap 1, Black stuck {{4 | 2} | {0 | −2}}
level material		 L One king, and two files to be in
gap 2, White may double + gap 3, White may double 3·↑∗
level material		 L A pawn ending is a sum
gap 3, Black stuck + gap 2, contested 4 | 2
level material		 L A position with no value, and the rule that gives it one
gap 3, contested + gap 2, White stuck + gap 1 {∗ | −4∗}
level material		 R What has to break before a pawn is worth a number
gap 3, White may double + gap 3, Black may double 0
level material		 P A pawn ending is a sum
gap 4, contested + gap 4, Black stuck 7 | 1
level material		 L What has to break before a pawn is worth a number

pawn file

17 positions

PositionWorthOutcomeDrawn in
gap 1 ∗
one file of a blocked ending		 N A pawn ending is a sum, What has to break before a pawn is worth a number
gap 1, Black may double ∗
both mobile, one double step		 N What has to break before a pawn is worth a number
gap 1, Black's pawn cannot advance 1
one file of a blocked ending		 L What has to break before a pawn is worth a number
gap 1, the square is contested, White's pawn cannot advance −1
contested, and one pawn already stopped		 R What has to break before a pawn is worth a number
gap 1, White's pawn cannot advance −1
one pawn cannot advance		 R What has to break before a pawn is worth a number
gap 2 0
one file of a blocked ending		 P A pawn ending is a sum
gap 2 (White still on the starting rank) ↑
one file of a blocked ending		 L A pawn ending is a sum
gap 2, Black's pawn cannot advance 2
one file of a blocked ending		 L One king, and two files to be in, What has to break before a pawn is worth a number
gap 2, the square is contested 1 | −1
one file of a blocked ending		 N One king, and two files to be in, What has to break before a pawn is worth a number
gap 3 ∗
one file of a blocked ending		 N A pawn ending is a sum
gap 3 (White still on the starting rank) 2·↑∗
one file of a blocked ending		 L A pawn ending is a sum
gap 3, Black's pawn cannot advance 3
one file of a blocked ending		 L What has to break before a pawn is worth a number
gap 3, the square is contested 2 | −2
one file of a blocked ending		 N What has to break before a pawn is worth a number
gap 3, White's pawn cannot advance −3
one file of a blocked ending		 R What has to break before a pawn is worth a number
gap 4 0
one file of a blocked ending		 P A pawn ending is a sum
gap 4 (White still on the starting rank) 3·↑
one file of a blocked ending		 L A pawn ending is a sum
gap 4, the square is contested 3 | −3
one file of a blocked ending		 N What has to break before a pawn is worth a number

subtraction of 1, 3, 4

13 positions

PositionWorthOutcomeDrawn in
heap 0 0
equal to that Nim heap in every sum, which is what the table checks		 P Every impartial game is a Nim heap
heap 1 ∗1
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 10 ∗1
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 11 ∗2
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 12 ∗3
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 2 0
equal to that Nim heap in every sum, which is what the table checks		 P Every impartial game is a Nim heap
heap 3 ∗1
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 4 ∗2
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 5 ∗3
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 6 ∗2
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 7 0
equal to that Nim heap in every sum, which is what the table checks		 P Every impartial game is a Nim heap
heap 8 ∗1
equal to that Nim heap in every sum, which is what the table checks		 N Every impartial game is a Nim heap
heap 9 0
equal to that Nim heap in every sum, which is what the table checks		 P Every impartial game is a Nim heap

the octal game ·007

9 positions

PositionWorthOutcomeDrawn in
the octal game ·007, a heap of 1 genus 0^120
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N The genus of a sum
the octal game ·007, a heap of 2 genus 0^120
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N The genus of a sum
the octal game ·007, a heap of 3 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P The genus of a sum
the octal game ·007, a heap of 4 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P The genus of a sum
the octal game ·007, a heap of 5 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P The genus of a sum
the octal game ·007, a heap of 6 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N The genus of a sum
the octal game ·007, a heap of 7 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N The genus of a sum
the octal game ·007, a heap of 8 genus 0^120
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N The genus of a sum
the octal game ·007, a heap of 9 genus 3^31
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N The genus of a sum

the octal game ·6

14 positions

PositionWorthOutcomeDrawn in
the octal game ·6, a heap of 1 genus 0^120
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, What a tame heap may be replaced by
the octal game ·6, a heap of 10 genus 3^1431
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
the octal game ·6, a heap of 11 genus 4^0564
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild
the octal game ·6, a heap of 12 genus 0^20
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
the octal game ·6, a heap of 13 genus 3^1431
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
the octal game ·6, a heap of 14 genus 4^0564
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild
the octal game ·6, a heap of 2 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild, What a tame heap may be replaced by
the octal game ·6, a heap of 3 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, What a tame heap may be replaced by
the octal game ·6, a heap of 4 genus 0^120
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, What a tame heap may be replaced by
the octal game ·6, a heap of 5 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild
the octal game ·6, a heap of 6 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
the octal game ·6, a heap of 7 genus 3^1431
wild — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
the octal game ·6, a heap of 8 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild
the octal game ·6, a heap of 9 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild

the subtraction game {1, 2}

12 positions

PositionWorthOutcomeDrawn in
the subtraction game {1, 2}, a heap of 1 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild, What a tame heap may be replaced by
the subtraction game {1, 2}, a heap of 10 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild
the subtraction game {1, 2}, a heap of 11 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
the subtraction game {1, 2}, a heap of 12 genus 0^120
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
the subtraction game {1, 2}, a heap of 2 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, What a tame heap may be replaced by
the subtraction game {1, 2}, a heap of 3 genus 0^120
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild, What a tame heap may be replaced by
the subtraction game {1, 2}, a heap of 4 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild, What a tame heap may be replaced by
the subtraction game {1, 2}, a heap of 5 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
the subtraction game {1, 2}, a heap of 6 genus 0^120
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
the subtraction game {1, 2}, a heap of 7 genus 1^031
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 P Tame and wild
the subtraction game {1, 2}, a heap of 8 genus 2^20
tame — as a Nim heap — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild
the subtraction game {1, 2}, a heap of 9 genus 0^120
tame — as a row of single counters — the superscript is the misère Grundy value with 0, 1, 2, … heaps of ∗2 added		 N Tame and wild

How to read the outcome column

P means the previous player wins — whoever must move, loses. N means the next player wins, whoever that is. L and R mean Left or Right wins whoever starts. Only three of the four are comparisons with zero, which is where the subject departs from arithmetic. Every value here is under the normal-play convention; misère play has outcomes but no values, which is why the misère rows carry a dash.

The positions that play back · The figure library · All essays