A symmetry that is not a pairing

A pairing, and the pairing tested the reflection against the half turn, found it sound, found it settling 40 positions of its own, found it worth twenty nodes in twenty-one thousand to a solver, and closed on the one symmetry left:

The rung above is the quarter turn. It is the one symmetry not tested here and the only one that could plausibly beat the half turn at the root, because a square board has it and a square board is where Cram is usually played. A quarter turn sends a horizontal domino to a vertical one, so its self-paired dominoes are a different object again — there are none, since no domino is its own image under a ninety-degree rotation — and the fixed-square clause is the whole of it.

The clause is not the whole of it. There are two more, both invisible on the three maps this ladder has used, and the quarter turn fails both.

The check, as proposed #

The check is easy to write and it does what the rung below expected. The free squares are invariant under the quarter turn, no square is fixed by it, and there are no self-paired dominoes because a quarter turn sends a horizontal domino to a vertical one.

It fires on 19 positions of the $4 \times 4$ board, on six of the $3 \times 4$, on three of the $2 \times 4$ — and on the empty $4 \times 4$, which is the case the rung below cared about, because a check that fires at the root collapses a whole search.

So the first question comes back yes. It is the second that does not.

It is worth being clear about what the check is a check for, because the whole anchor turns on it. Cram is impartial and the question about a position is only who wins. A pairing check says: the player to move loses, because their opponent has a strategy that always has an answer. That is a strong claim to make about a position without searching it, which is why the check is worth putting in front of a solver at all, and why an unsound one is worse than none — the check that was not a check is the rung where an unsound version was found and repaired.

So a check contributes when it fires on a position a solver would otherwise expand. Firing on positions another check already settles contributes nothing, however often it happens.

And it settles nothing #

Every position the quarter-turn check fires on is a position the half-turn check fires on, on every board swept. The column of positions it settles alone reads nought, nought, nought, nought, nought.

That is not a coincidence of small boards, and the reason is one line of group theory. The square of a quarter turn is the half turn, so a set invariant under the quarter turn is invariant under the half turn; and on a square bounding box the only square either map can fix is the centre, so no fixed square means the same thing for both. Every quarter-turn firing is therefore a half-turn firing by construction.

A solver carrying the quarter-turn check would expand exactly the nodes it expands without it. There is nothing to measure.

What a pairing strategy actually needs #

The interesting question is why. The check passes all three clauses this ladder has been using for four rungs, and it is worthless — so the clauses are not the whole specification.

A pairing strategy is a repeatable answer: the opponent plays $D$, the strategy answers $\rho(D)$, the opponent plays $E$, the strategy answers $\rho(E)$, and so on until the opponent runs out. For that to work, the position after each pair of moves has to be $\rho$-symmetric again, or the strategy has nothing to appeal to on its next turn.

For the half turn and the two axis reflections that is automatic, because each map is its own inverse. Removing $D$ and $\rho(D)$ removes a $\rho$-orbit, and a $\rho$-symmetric set minus a $\rho$-orbit is $\rho$-symmetric.

A quarter turn has order four. Removing $D$ and $\rho(D)$ removes half an orbit, and what is left is symmetric under $\rho^2$ — the half turn — and not under $\rho$.

Measured on the empty $4 \times 4$: 24 first moves, and the quarter-turn reply restores the quarter-turn condition on none of them. The half-turn reply restores its own condition on all 24.

And sometimes there is no reply at all #

There is a second failure and it is more elementary.

On four of the 24 first moves the image of the move overlaps the move. A domino at the centre of a $4 \times 4$ board maps under the quarter turn to a domino sharing one of its squares, so the strategy’s reply is not a legal move — one of its two squares has just been covered.

The rung below checked the clause no domino is its own image, which is true and which is a different statement. For an involution the two clauses coincide: if $\rho(D) \neq D$ and $\rho$ is its own inverse, then $D$ and $\rho(D)$ are disjoint or they share a square, and sharing a square would make $\rho$ map one of $D$’s squares outside $D$ and the other inside, which forces a fixed square. For a quarter turn nothing of the kind follows, and the four central dominoes are the proof.

Why the two failures are one failure #

The illegal reply and the broken symmetry look like two separate defects and they are the same one seen from two distances.

A pairing is a partition of the free squares into pairs. An involution with no fixed point is such a partition — each square with its image — and everything a copying strategy needs follows from that: the reply is a legal domino because its two squares are the partners of two squares just covered and are therefore free; and the position afterwards is the same partition minus two pairs, which is still a partition.

An order-four map is not a partition of the squares into pairs. It partitions them into orbits of four, and there is no canonical way to read four squares as two pairs. So the strategy has no partner to play to; what it has is a next square, and following it round the orbit is a different argument that would need the opponent to cooperate.

Seen that way, the four central dominoes and the twenty-four broken symmetries are two symptoms of the same absence, and the clause the ladder should have been checking all along is a single one: the map is a fixed-point-free involution. Every other clause is a consequence.

Three symmetries, and why exactly three #

A square board has eight symmetries. This ladder has used three of them: the half turn and the two axis reflections. Nobody had said why those three.

The diagonal reflections are excluded for a reason that is easy to see and is not the one that matters here: they swap rows with columns, so they send a horizontal domino to a vertical one and the copying strategy’s reply is a different kind of move. Cram is impartial, so both players have both kinds and that is not immediately fatal — but a diagonal reflection fixes every square on its axis, so the fixed-square clause kills it outright.

The quarter turn and its inverse are excluded for the reason this page is about: they are not involutions.

So the three usable symmetries are exactly the three elements of order two that fix no square, and the property that decides it is that the map is its own inverse. That was invisible for four rungs because every map anyone tried had it.

What the empty board actually shows #

The rung below’s hope rested on one observation — a square board has a quarter turn — and it is worth following that observation to its end, because it is right and it does not help.

The empty $4 \times 4$ board is quarter-turn symmetric and has no fixed square, so the check fires there. It is also half-turn symmetric with no fixed square, so the half-turn check fires there too. Both are correct: the $4 \times 4$ Cram board is a second-player win, and the half-turn strategy is the standard proof of it.

What the rung below was hoping for was a board where the quarter turn fires and the half turn does not. For that the half turn would have to be blocked, and the only clause that can block it on a symmetric board is the self-paired domino — a domino through the centre, mapped to itself by the half turn. On an even-sided board there is no centre square, so no domino is half-turn invariant, and the half turn is never blocked; on an odd-sided board the centre square is fixed by both maps, so neither check fires. There is no gap for the quarter turn to fill, and the census finding it empty is the arithmetic rather than the sample.

What this is an instance of #

The general shape is worth naming, because it is not a fact about Cram.

A copying strategy is an argument that the second player can always answer. Its content is that the position has a symmetry the answer preserves, and preservation is what makes the argument inductive rather than a statement about one move. The clauses a ladder ends up checking — every square has a partner, no square is its own, no move is its own image — are the consequences of that on the maps it happens to use, and a consequence is not a definition.

The strategy that is a symmetry is where the copying argument arrives on this site, and it states the requirement correctly: the answer restores the position’s symmetry. Four rungs of measurement then narrowed the check to three clauses that are equivalent to it for involutions, and the narrowing was invisible because it was never wrong.

That is the ordinary way a working criterion drifts from the thing it was a criterion for. It stops being equivalent at exactly the first case outside the family it was calibrated on, and the case is unusually clean here: the quarter turn passes every clause and fails the requirement.

What this does not settle #

The boards are small. Five boards, the largest $4 \times 4$, with every position reachable by play. The subset argument — quarter-invariance implies half-turn invariance — does not depend on the board, so nothing about the census is expected to change; what a larger board would add is more firings of a check already known to contribute nothing.

Only Cram. The whole anchor is about Cram, which is impartial and whose dominoes are the same for both players. In Domineering a quarter turn would fail earlier and more obviously, because it sends a Left move to a Right move; the interesting content here is that Cram removes that objection and the quarter turn still fails.

The reply test is one board. Twenty-four first moves on the empty $4 \times 4$, which is the position the rung below named and the smallest square board on which the quarter-turn check fires at all. The failure is structural — an order-four map cannot restore its own symmetry after one application — so nothing about a larger board would change it, and nothing about a larger board has been checked.

And the two new clauses are not a complete specification either. The reply is legal and the reply restores the symmetry are the two failures this page found, and they are consequences of the same underlying requirement rather than a list. The correct statement is the one the strategy that is a symmetry already makes; what this page adds is a case where the convenient restatement of it is wrong.

The quarter turn is sound where it fires, and that is not a defence of it. It is sound because it fires only where the half turn does, and the half turn is sound — so its soundness is inherited rather than earned, and a check that never fires alone can never be caught being wrong.

Nothing here is drawn. Every figure is a count, and the argument’s two failures are both about a picture: a domino overlapping its own image, and a board that stops being symmetric after two moves. Both are drawable in a way almost nothing else on this anchor is — four squares at the centre of a $4 \times 4$ grid would carry the first entirely — and the tables carry the counts instead, because a drawing of the four central dominoes is a drawing of one board and the claim is about every board with a square bounding box. Looking for the symmetry is where the shapes get drawn on this anchor.

Normal play throughout. Under misère Cram the copying strategy inverts, as it does everywhere: the player who always has an answer is the player who has to make the last move.

What is left of the rung below’s hope #

Four rungs of this anchor have been asking whether the half turn is the pairing or a pairing, and the answer is now nearly complete.

The reflections are genuine alternatives: sound, contributing 40 positions of their own on a $4 \times 5$ board, and worth about one node in a thousand to a solver. The quarter turn is not an alternative at all. The diagonals are not either. So of the eight symmetries a square board has, three carry a pairing argument, one of them is the half turn, and the other two are worth almost nothing because they cannot fire on an empty rectangle.

That leaves the ladder’s subject as the half turn for a reason that is now fully stated: it is the only fixed-point-free involution among a rectangle’s symmetries that an empty rectangle has, and a pairing check is worth most exactly where a search has only one node to save. Both halves of that sentence were measured on separate rungs and this is the first page where they meet.

Where the ladder goes next #

The pairing anchor has seven rungs: the strategy that is a symmetry, looking for the symmetry, the symmetry one move away, a check in front of a search, the check that was not a check, the other pairings, and now the symmetry that is not one.

The rung above is the search for pairings that are not symmetries of the board at all. Every map this ladder has tried is a rigid motion of the rectangle, and the requirement — an involution on the free squares, with no fixed square and no domino meeting its own image — mentions nothing of the kind. A position could have a pairing that is not geometric: an arbitrary involution on its free squares, found by search rather than read off the shape. Looking for the symmetry already searches shapes for a half-turn axis; searching a shape for any involution meeting the three clauses is a much larger search over a much larger space, and it would say whether the geometric pairings are the only ones a Cram position has or merely the only ones anybody has looked for.

Two neighbours are worth the trip. A check in front of a search is where a check’s worth was priced as firing rate times subtree saved, and it is the formula that makes this page’s answer a nought rather than a small number. And the check that was not a check is where the half turn’s own clause list was repaired, and it is worth reading beside a page about a clause list being incomplete in the other direction.