The capture that has to be made

The convention Dawson actually used ended with an admission in its list of limits. Every computation on that page and on the two before it is about ·137, a row of counters with a three-digit rule. None of them is about pawns. The step from a three-rank pawn diagram to a row of counters, it said, is a reduction taken on trust from the survey literature rather than performed.

A reduction taken on trust is a claim that two different games are one game. Both games are small enough here to search outright, so the claim can be checked, and checking it turns out to say more than confirming it. The pawns do turn into counters. They do so because of one sentence of the diagram’s rules that has no counterpart anywhere in the three digits of ·137.

The diagram #

The board is three ranks deep and some number of files wide. A White pawn stands on every square of the first rank and a Black pawn on every square of the third; the middle rank is empty. A pawn moves as in chess, one square straight ahead into an empty square, or one square diagonally ahead to capture an enemy pawn. Two rules are Dawson’s rather than chess’s. There are no other pieces, and a capture, when one is available to the player to move, must be made.

The ending is the third rule, and it is the one the earlier essay was about. Dawson posed his problem with the player who cannot move winning. The modern octal game is usually quoted with that player losing. Both are searched here.

One step, and four captures nobody may decline #

Watch a single move on five files. White steps the middle pawn forward. It now stands diagonally in front of two Black pawns, and Black, obliged to capture, takes it with one of them. That Black pawn stands diagonally in front of two White pawns, and White must retake. The retaking pawn is attacked by the other Black pawn beside it, which must take in turn, and the last White pawn beside it must retake. Five moves after the step, the compulsion runs out.

What is left is a finished block. On the middle file a White pawn faces a Black pawn head on, and neither can move or capture. On the two files beside it all three squares are empty. The outer two files are exactly as they began, each a White pawn and a Black pawn with an empty square between. And it is Black to move, as it would be after any one move of White’s.

That is a move of ·137 in plain sight. A heap of five counters has lost three from its middle and become two heaps of one, and the turn has passed once. The five moves of chess were one move of the counter game, because four of them were not choices.

Every exchange is a move of ·137 #

One exchange is an illustration. The reduction is a claim about all of them, so all of them are replayed. From every first move on every board from one file to twelve — 78 first moves in all — the forced captures are followed until the player to move has none, along every branch where there is a choice of which pawn captures.

Three shapes come out, and they are the three digits of ·137. A pawn alone on a single file steps forward and is blocked, one move, and a heap of one is gone: the 1, which permits taking a single counter only when it is the whole heap. A pawn at the edge of a longer row steps forward, and the exchange it starts lasts three moves in all, finishing two files and leaving the rest of the row as one heap: the 3, which permits taking two counters and leaving nothing or one heap. And a pawn anywhere else starts the five-move exchange above, finishing three files and leaving nothing, one heap or two: the 7.

Two checks come with the replay and both are what make it a proof on these boards rather than a picture. Every exchange lasts an odd number of moves, so the player who starts an exchange is always the player who hands the turn over at its end — the property that lets a chess move count as a counter move under either ending. And where two pawns could capture, both choices end in the same files, so the exchange has one outcome however it is played. Across the 78 first moves the files left are always exactly one of ·137’s options from a heap of that size, and every one of those options is reached.

Twelve boards against the sequence #

The exchange check says the two games have the same moves. The outcome check asks the question the essays actually quote, and asks it of the chess position directly: the search knows nothing about heaps, and simply plays pawns until somebody is stuck.

On every board from one file to twelve, the first player wins the pawn diagram exactly when ·137’s Grundy value is not nought — four files and eight are the second player’s, every other size the first player’s. And under Dawson’s own ending, where the stuck player wins, the pawn diagram agrees with ·137’s misère outcome on every one of the twelve.

The second agreement is the one worth pausing on, because the earlier essay found the two endings producing objects of different kinds. Under the last-move-wins rule a heap carries a nimber, and under Dawson’s rule it carries a genus and a class that grows with the heaps in view. The reduction does not care. It maps a chess move to a counter move and a chess turn to a counter turn, and it preserves who moves last, so whatever either ending makes of ·137 it makes of the pawns. That is the odd length of every exchange doing its work, and it is what licenses the earlier essay’s treatment of Dawson’s convention as a statement about his pawns at all.

What the pawns are worth #

With the reduction checked, the sequence a chess problem that was an octal game drew becomes a statement about pawns that can be read directly. A board of four files is a loss for the player to move because ·137’s value at four is nought; a board of five is worth ∗3; two diagrams side by side on one wide board, separated by a finished block, are worth the exclusive-or of their values.

The search reaches twelve files, and the values past that are the sequence’s rather than the search’s. A reader who wants to know who wins the diagram on fifty files is reading ·137 at fifty and trusting that the exchanges keep behaving as they did on twelve. The structure of an exchange gives good reason for that trust — every exchange is local, touches at most three files, and depends on nothing further along the row — but that is an argument, and the checks above are measurements. What the arithmetic cost in 1956 priced the sequence; nothing on this page extends the pawn search to meet it.

The same diagram with captures optional #

The octal code contains no word about compulsion, so the natural question is whether the compulsion matters or whether it is a problemist’s decoration the reduction would survive without.

It matters completely. With captures optional, the winner changes on boards of two, three, six and seven files under the rule that the last move wins, and on two, six and seven under Dawson’s. On one, four and five files the winners happen to coincide, and they coincide for no reason the reduction supplies: the optional game is no longer ·137, and agreement on a board is an accident of that board.

The positions show why it is a different game rather than a variant. With captures compulsory the pawn search on seven files visits 210 positions under the two endings together, because an exchange, once started, has at most one or two ways to finish. With captures optional it visits 1,168,860, because every exchange can now stop after any of its moves, and a half-finished exchange — a pawn standing in front of two enemies that have declined to take it — is a position with a character of its own. An exchange that nobody has to finish is not a move of anything.

The optional game also plays on squares the compulsory one never touches. A White pawn that has stepped forward and not been taken can capture onto Black’s home rank, and a Black pawn left alone can do the same in the other direction. With captures compulsory that never happens: on every board from one file to twelve, no position the search visits has a pawn on its far rank, because the pawn that could capture there is always taken first. With captures optional it is nearly the whole game. At seven files, 1,154,644 of the 1,168,860 positions the search visits have a pawn standing on its far rank; at six, 97,809 of 101,387; at four, 667 of 847.

That count changes what the optional results mean, and it has to be said plainly. The model gives a pawn on its far rank no move, where chess would promote it. In the compulsory game the rule is never consulted, so the reduction to ·137 owes it nothing. In the optional game it is consulted almost everywhere, so the optional winners in the table are winners of a game in which a pawn that breaks through becomes inert — a defensible model of the diagram, and not chess. Promoted pieces would add moves on almost every position the optional search visits, and nothing here says which of the optional winners would survive them.

Two games on one board #

The far-rank count also says something about the shape of the two games that the winners alone do not.

The compulsory game is a game of blocked pawns. Every exchange ends with a White pawn and a Black pawn face to face on one file and the files beside it emptied, and no pawn in any position it reaches stands further forward than the middle rank. Its positions are a row of untouched files, a scattering of finished blocks, and at most one exchange in progress. That is why its search stays so small — 2,301 positions at twelve files under both endings together, against the 1,168,860 the optional game needs at seven — and it is the same fact ·137 expresses by needing only one number per heap. The pawn game has almost nothing in it that the counters do not.

The optional game is a game of pawns breaking through. Once a capture may be declined, a pawn that has stepped forward can be left standing, the pawns beside it can step past one another, and almost every position reachable from the start has a pawn on the far side of the board. The two games begin from the same diagram and share the same first move, and after a handful of moves they occupy almost disjoint parts of the space of pawn arrangements. The octal game is a faithful description of the first region and says nothing about the second.

That is why the smallest disagreement is worth drawing in full, and why it is the right witness rather than the larger ones. On two files the whole game is three positions long, and no pawn in it ever reaches a far rank: the line ends with both White pawns blocked on the first two ranks. So the two-file result does not depend on the promotion rule at all. It is the compulsion alone — Black allowed to leave a pawn standing — that turns ·137’s first-player win into a second-player win. The disagreements on three, six and seven files are real disagreements in this model, and they are not shown to be free of the promotion choice.

A capture declined #

The smallest board where the compulsion decides the winner has two files, and the whole game fits in three diagrams. White steps a pawn forward. Under Dawson’s rule Black must take it, White retakes, and White has made the last move: the first player wins, as ·137’s value of ∗1 at two says.

With captures optional Black does something the rule forbade. It leaves the advanced pawn where it is and steps its other pawn forward instead, beside it. Now the advanced White pawn is blocked by the Black pawn in front of it and has nothing diagonally ahead to capture; the other White pawn is blocked by the Black pawn that just arrived. White is to move and has no move. The capture Black declined was the move that would have lost.

That is the reduction’s hinge seen from the other side. Compulsory capture is what keeps a pawn from being a threat that can be left standing, and a threat left standing is precisely the kind of position a row of counters has no way to express.

Where the reduction’s weight sits #

A pawn ending is a sum found the same vocabulary in another corner of chess, where a blocked pawn ending decomposes into files that never touch and a zugzwang is a P-position. There the independence of the parts came from the pawns being blocked. Here it comes from something less visible: each exchange is forced to completion, so no interaction between neighbouring files survives the move that starts it.

In both cases the translation into this subject’s vocabulary rests on a clause of the move rule rather than on anything about the material. What has to break before a pawn is worth a number measured how the values of a pawn file change when one clause is broken, and this is the same kind of measurement made on a different clause. Three bits of rule found that a single bit of an octal code — whether a move may split a heap — separates the codes that settle from those that might not. The compulsion here plays that role one level up: it is the rule that makes the diagram an octal game at all.

What the searches cannot show #

Twelve files and seven. The compulsory diagram is searched to twelve files and the optional one to seven; past those, the claims are the sequence’s. The argument that exchanges stay local is sound, and it is an argument, not a search.

The diagram is the standard one. Every board searched starts full, with every file live. Dawson’s own problem, as the earlier essay records, was a particular position, and nothing here reconstructs which one; the reduction checked is the one the octal game is quoted for.

Promotion is set aside. A pawn on its far rank has no move in this model. That costs nothing in the compulsory game, which never produces such a pawn on any board searched, and it shapes nearly every position of the optional one — at seven files, all but 14,216 of the 1,168,860. The optional winners are therefore winners of that model. Only the two-file disagreement is independent of the choice, because no pawn in its line reaches a far rank.

And chess has kings. Dawson’s diagram has none, and a king on the board would couple files together in exactly the way one king, and two files to be in measured for blocked pawn endings.

The rules the counts assume #

A board three ranks deep and n files wide, White pawns on the first rank and Black pawns on the third. A pawn steps straight ahead into an empty square or captures diagonally ahead onto an enemy pawn; a pawn on the far rank has no move. With captures compulsory, a player who has a capture available must make one. White moves first. Under the ending ·137 is usually quoted with, normal play, the player with no move loses; under Dawson’s, misère play, that player wins. ·137 is the octal game whose digits permit taking one counter only as a whole heap, two counters leaving at most one heap, and three counters leaving at most two.

Still open: a diagram the counters cannot hold #

The optional game is not ·137, and it is not obviously any octal game. A half-finished exchange is a position in which one file’s pawn threatens its neighbours, and whether such a position can be written as a sum of independent parts at all — or whether the optional diagram is a game in which files genuinely interact — is not settled here. The measurement that would begin to settle it is the one a pawn ending is a sum makes for blocked files: compute the optional diagram on two separated rows of files, and ask whether its outcome is a function of the two rows’ outcomes alone.