Generator

Clobber: every value smaller than every number

Blue and red stones on a small board. A move takes one of your own stones onto an orthogonally adjacent enemy stone, which is removed. Because adjacency is symmetric, a player has a move exactly when the opponent does — so no position can ever be worth a whole move to anybody, and every value that comes out is an infinitesimal.

5 essays call clobber-board. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

The positions it draws

34 distinct positions, harvested by running this generator again at the options each essay passed it.

Position	Worth	Outcome	Drawn 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

Where it is called

Changing this generator changes every one of these figures.

Particular games

A game where nobody can be ahead in moves

A blue stone beside a red one is a move for both players at once. So neither player can run out while the other still has something to do — and every value the game produces is smaller than every positive number, by the shape of the rule rather than by inspection.

Values

The class where nobody runs out first

Three stones in a row — blue, blue, red — and the position is worth exactly up. Clobber cannot produce anything else, because adjacency is symmetric — a player has a move precisely when the opponent does, and a game with that shape can never be worth a whole move to anybody.

The bracket of a sum, against the sum of the brackets. Two all-small positions, the interval of multiples of ↑ each lies between, those two intervals added coordinatewise, and the interval the sum actually lies between. The added one always contains the computed one — greater-than survives addition — so the bracket never widens under a sum. Where it narrows, the parts were each too vague to pin down and the sum is not.

Sums and comparison

When the ups add

Atomic weight brackets do not add over a sum — they bound it. Over all 120 pairs from a fifteen-game family the sum's bracket came out exactly the sum of the parts' brackets 56 times, strictly narrower 64 times, and wider never; and the rule separating the two is one line long, because every one of the 54 pairs with a pinned part is exact and only 2 of the other 66 are.

Particular games

One row of Clobber

Every string of blue, red and empty squares up to eight long — 9,840 rows — carries one of only 111 values, and every one of them is infinitesimal. A third of the rows are worth exactly zero. Six alternating stones are worth zero and eight are worth a form that takes four lines to print, so the values do not simplify as the row grows: they explode, while the row stays trivial to describe.

What an approximation is worth. Every pair of Clobber rows up to six squares, judged twice: by their up-brackets and by the comparison itself. The bracket is never wrong where it speaks, and most of what it declines to answer has no answer.

Particular games

When the bracket decides

A Clobber row's value is an all-small game nobody can hold in their head, so the practical answer is the up-bracket: a pair of integers between which its atomic weight must lie. As an approximation it is worth exactly what it settles — 1,585 of 7,875 pairs of rows are ordered by it, every one of those orders is right, and of the 6,290 it declines, 4,222 have no answer either.

The whole library · The position index · The figures that play back