Combinatorial Game Theory
who wins, and by how much
Generator

A blocked file, and the tempo it holds

A blocked file, and the tempo it holds
A blocked file, and the tempo it holds. Files of a blocked pawn ending: a White pawn below, a Black pawn above, and a gap between them that either side may close one square at a time. Each file carries the value the game recursion gives it. With only single steps available a file is worth a star or nothing, by the parity of the gap, so the whole position is tempo and no material at all — which is what a chess player means by mutual zugzwang.

Files of a blocked pawn ending: a White pawn below, a Black pawn above, and a gap between them that either side may close one square at a time. Each file carries the value the game recursion gives it. With only single steps available a file is worth a star or nothing, by the parity of the gap, so the whole position is tempo and no material at all — which is what a chess player means by mutual zugzwang.

4 essays call pawn-file. 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

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

PositionWorth OutcomeDrawn in
gap 1 + gap 1 0 P A pawn ending is a sum
gap 1 + gap 2 + gap 3 0 P A pawn ending is a sum
gap 1 + gap 3, White may double ⇑ L A pawn ending is a sum
gap 1, contested + gap 1, contested 0 P One king, and two files to be in
gap 2, contested + gap 1 {1∗ | −1∗} N A position with no value, and the rule that gives it one
gap 2, contested + gap 1, Black stuck 2 | 0 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}} L One king, and two files to be in
gap 2, White may double + gap 3, White may double 3·↑∗ L A pawn ending is a sum
gap 3, Black stuck + gap 2, contested 4 | 2 L A position with no value, and the rule that gives it one
gap 3, contested + gap 2, White stuck + gap 1 {∗ | −4∗} R What has to break before a pawn is worth a number
gap 3, White may double + gap 3, Black may double 0 P A pawn ending is a sum
gap 4, contested + gap 4, Black stuck 7 | 1 L What has to break before a pawn is worth a number
gap 1 ∗ N A pawn ending is a sum · What has to break before a pawn is worth a number
gap 1, Black may double ∗ N What has to break before a pawn is worth a number
gap 1, Black's pawn cannot advance 1 L What has to break before a pawn is worth a number
gap 1, the square is contested, White's pawn cannot advance −1 R What has to break before a pawn is worth a number
gap 1, White's pawn cannot advance −1 R What has to break before a pawn is worth a number
gap 2 0 P A pawn ending is a sum
gap 2 (White still on the starting rank) ↑ L A pawn ending is a sum
gap 2, Black's pawn cannot advance 2 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 N One king, and two files to be in · What has to break before a pawn is worth a number
gap 3 ∗ N A pawn ending is a sum
gap 3 (White still on the starting rank) 2·↑∗ L A pawn ending is a sum
gap 3, Black's pawn cannot advance 3 L What has to break before a pawn is worth a number
gap 3, the square is contested 2 | −2 N What has to break before a pawn is worth a number
gap 3, White's pawn cannot advance −3 R What has to break before a pawn is worth a number
gap 4 0 P A pawn ending is a sum
gap 4 (White still on the starting rank) 3·↑ L A pawn ending is a sum
gap 4, the square is contested 3 | −3 N What has to break before a pawn is worth a number

Where it is called

Changing this generator changes every one of these figures.

A blocked file, and the tempo it holds. Files of a blocked pawn ending: a White pawn below, a Black pawn above, and a gap between them that either side may close one square at a time. Each file carries the value the game recursion gives it. With only single steps available a file is worth a star or nothing, by the parity of the gap, so the whole position is tempo and no material at all — which is what a chess player means by mutual zugzwang. Out in the world

A pawn ending is a sum

In a blocked pawn ending the material is level, the files never speak to each other, and whoever has to move is the one in trouble. Chess calls that mutual zugzwang; this site calls it a P-position; and the two vocabularies were built four decades and one subject apart to say the same thing.

 Which clause of the rules produces which kind of value. Every combination of pawn-file clause in range, sorted by the kind of value it produces. Files where both pawns can advance are all-small and their values are nimbers and infinitesimals. A file where one pawn is stuck behind a friendly piece gives the other side free moves and is worth an integer. A file whose middle square can be held stops the other pawn the moment somebody reaches it, and is worth a switch — a position both players want to move in. The dictionary is read off the evaluation rather than asserted. Out in the world

What has to break before a pawn is worth a number

Every value the blocked-file model can hold is an infinitesimal, and the reason is one sentence about the move rule rather than anything about pawns. Break that sentence — a pawn stuck behind a friend, a square only one side can hold — and integers, switches and positions worth fighting over arrive at once.

 What one king costs a decomposition. Two pawn files and one king a side, solved as a joint position and again as the sum of its files. With the kings unable to move the two answers agree on every configuration, because a king that cannot choose between files is not a shared resource. Give each king a waiting move and the answers come apart, and on some configurations the sum of the parts names the wrong winner rather than merely the wrong value. Independence is a hypothesis about the position and this is the price of assuming it wrongly. Out in the world

One king, and two files to be in

The whole apparatus needs the files to be independent, and a king is what makes them not. With the kings unable to move the sum of the parts is exact on every configuration; give each king a single waiting move and the sum names the wrong winner on one configuration in six, and on a hundred and twenty-six of two hundred and forty-three with three files.

 A fortress, and the counter that gives it a label. A pawn ending where the defender's king shuffles for ever and the attacker needs time. Down the rows, how many moves of preparation the breakthrough needs; across the columns, how many moves the rule allows before declaring a draw. With no breakthrough the position is drawn whatever the rule says, and drawn as a residue the backward induction never reaches. With a breakthrough and no rule the attacker wins despite the cycle. Where the march is longer than the counter allows, the rule turns a won position into a drawn one. Out in the world

A position with no value, and the rule that gives it one

A fortress is a cycle in the position graph, so the recursion defining a value has nowhere to bottom out and the propagation never reaches it. Chess has a rule for that — count fifty moves and call it drawn — and the rule does not merely tidy the theory up. On eleven cells of the sweep it takes away a win.

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