Combinatorial Game Theory
who wins, and by how much
Generator

Which part to move in — the which part generator

Which part to move in
Which part to move in. A sum, and every move one player has in it. Each row is a component, the option taken in it, and what the whole position becomes. The values of the parts say who wins; they do not say where to play, and the winning move here is in the component worth the least.

A sum, and every move one player has in it. Each row is a component, the option taken in it, and what the whole position becomes. The values of the parts say who wins; they do not say where to play, and the winning move here is in the component worth the least.

6 essays call which-part. 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

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

PositionWorth OutcomeDrawn 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
{∗ | ↓} {∗ | ↓} R The class where nobody runs out first · When the ups add
{∗ | ↓} + ↑ {↑∗ | 0} R The class where nobody runs out first · When the ups add
{0 | -2} 0 | −2 N Which part to move in
{1 | 0} 1 | 0 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} + 1/2 + ∗ + ↑ + 0 {3/2↑∗ | 1/2↑∗} L What is left when the small change is thrown away
{2 | 0} 2 | 0 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} + {1 | 0} {{3 | 2} | {1 | 0}} L Numbers avoid numbers
{2 | 0} + 1 3 | 1 L Numbers avoid numbers
{3 | 1} 3 | 1 L The endgame, accounted for
{4 | 0} 4 | 0 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
↑ ↑ 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
↑ + ∗ ↑∗ 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
∗ ∗ 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
∗ + ∗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
∗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
∗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
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
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 + 1/2 3/2 L Numbers avoid numbers
1/2 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
green E ∗ N The sum is the object · Which part to move in
green EE ∗2 N Which part to move in
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

Where it is called

Changing this generator changes every one of these figures.

Which part to move in. A sum, and every move one player has in it. Each row is a component, the option taken in it, and what the whole position becomes. The values of the parts say who wins; they do not say where to play, and the winning move here is in the component worth the least. Sums and comparison

Which part to move in

The value of a sum is the sum of the values. The move in a sum is not the move in any part, and there is no rule that reads it off the values — in the smallest interesting example, the only winning move is in the component worth nothing.

 The endgame, accounted for. Several independent regions, each a fight with a settled value and a size. The account plays them hottest first: add up what each is worth on average, then add the largest amount at stake, subtract the next, and so on down. The exact value of the whole position is computed beside it, and the figure prints both. Temperature

The endgame, accounted for

Add up what each region is worth, then add the biggest thing at stake, subtract the next, and so on down. On a board of simple fights the result is exact — and the moment one region has a fight inside it, the account is out by a point.

 Why nobody moves in the number. A hot position added to a number. Left wins the sum whoever moves — but only by moving in the fight. Spending the move on the number instead hands the position back as a first-player win, with Right to move, which throws the win away. The theorem says this is always so, and here it is happening. Values

Numbers avoid numbers

In a position with a number in it and anything else, the number is never the right move. That is a theorem rather than a heuristic, and it is the closest this subject comes to advice a player can carry into a real game.

 Three ways to add the same games. One list of components, added three different ways. Under the disjunctive rule a move is a move in exactly one part; under the conjunctive rule it is a move in every part at once, and play stops as soon as any part runs out; under the selective rule it is a move in any non-empty set of parts. The outcomes are computed by search from each rule's own definition. Sums and comparison

Three ways to add the same games

A move in exactly one component is a choice, not a law. Move in every component at once and the game is different; move in any set of them and it is different again. The same two positions, added three ways, give three different answers — and only one of the three has values that add.

 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. 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.

 What the reduction collapses. Each reduced form with the values that reduce to it. The largest class is the one that reduces to zero and it holds every infinitesimal on the list, which is exactly what the reduction is for — against a hot background, none of them is distinguishable from nothing. Sums and comparison

What is left when the small change is thrown away

Canonical form answers a demanding question: which positions are interchangeable inside every sum whatever. A player with a hot board does not have every sum — an infinitesimal difference cannot decide anything against a genuine fight — so there is a coarser question with an exact answer. The reduced canonical form takes the 1,474 values born by day three to 61, with 292 of them collapsing to zero, and it is a homomorphism on all 8,100 pairs tested only when a second pass is made.

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