Every move closes the largest gap

A parity with a first exception solves every Sylver Coinage position with up to fifteen numbers left unnameable and finds the parity of that count very nearly deciding the game. It ends by making the next question precise without answering it. The positions are numerical semigroups, and numerical semigroups are studied for reasons that have nothing to do with games — their counts by genus, the shape of their gap sets, which of them are symmetric — so every one of those properties can now be asked against the outcome column.

That question has a clean answer for exactly one property, and the property is the one the list ends on. A symmetric numerical semigroup is never a lost position, and neither is its near relation, the pseudo-symmetric one. The census below finds one exception in 584, and the exception is the smallest position there is.

And the reason is not a pattern. It is an argument already met twice, on Hex and on Chomp.

Gaps that pair off #

A Sylver Coinage position is a numerical semigroup: the numbers reachable as sums of what has been named. Its gaps are the numbers still nameable, and its Frobenius number F is the largest gap. The genus is how many gaps there are.

A semigroup is symmetric when its gaps pair off around F: for every gap x, the number F − x is not a gap — it is already reachable. Since x and F − x cannot both be gaps and cannot both be reachable, the numbers from 0 to F split exactly in half, and a symmetric semigroup has genus (F + 1)/2.

Every semigroup with two generators is symmetric, and the strip shows why the pairing matters for the game rather than only for the algebra. Take any gap x and name it. Its partner F − x is already reachable, so x + (F − x) = F is now reachable too. Naming any gap of a symmetric position closes the largest gap as well.

A pseudo-symmetric semigroup is the same with one exception: F is even, and the single gap F/2 has no partner, since F − F/2 is itself. But naming F/2 closes F anyway, because F/2 + F/2 = F. So in a pseudo-symmetric position too, every move closes the largest gap.

Together the symmetric and pseudo-symmetric semigroups are called irreducible, for a reason from the theory of semigroups that is not needed here. What is needed is the property the two strips share: every move from the position makes F nameable.

One lost position in 584 #

The census is the one the earlier essay ran, extended by a genus, and its outcome column comes from the same recursion on the semigroups themselves. Beside it are two tests that know nothing about the game: whether the gaps pair off around F, and whether they do except at F/2.

Of 584 irreducible positions, one is lost, and it is the position at genus one. Its only gap is 1; naming 1 is the move nobody makes voluntarily, and here it is the only move there is. Every other symmetric or pseudo-symmetric position, at every genus from two to sixteen, is a win for the player to move.

The rest of the census could hardly be more different. Among the 11,185 positions that are not irreducible there are 1,405 losses — about one in eight — and at odd genus the share is far higher than that, as the earlier essay’s parity table says. So the split is not a refinement of the parity. At genus fifteen, the largest odd row, 615 of the 2,737 ordinary positions are lost and none of the 120 irreducible ones is.

Why: the move every other move contains #

The shape of the argument will be familiar from the theorem that names no move, and the fit is exact.

Suppose the player to move names F. Either that leaves a position the opponent loses, in which case the mover has won, or it does not — and then the opponent has a winning reply: some number y whose naming leaves the mover lost.

Now go back to the original position and name y first. Every move from an irreducible position closes F, so naming y closes F as well: the reachable set after naming y is the reachable set after naming F and then y. The mover has reached, in one move, exactly the position the opponent’s winning reply would have reached — with the opponent to move. And that position is lost for whoever is to move in it. So the mover wins.

Either way the mover wins, which is all a stealing argument ever delivers. It fails only if naming F is not a real option — and the one position where it is not is the position whose Frobenius number is 1, since naming 1 is the move that loses on the spot. That is precisely the single exception in the census.

All 7,736 moves close the largest gap, which is the hypothesis, checked rather than argued. And then the part the argument does not supply: the move it talks about wins on only 123 of the 583 positions. On the semigroup of 3 and 7, naming 11 loses and the only winning move is 2.

The argument, run on one position #

The semigroup of 3 and 7 is small enough to watch the argument happen rather than trust it, and it happens exactly as described.

Name 11, the move the argument is about. What is left has gaps 1, 2, 4, 5 and 8, and the opponent, now to move, wins — by naming 2, which closes every remaining gap except 1 and leaves the original mover with nothing but the losing move.

The argument’s second step says the mover could have played that reply first. And so it could: 2 was a gap of the semigroup of 3 and 7 all along. Named at once, 2 closes 4 and 5 and 8 — and 11 as well, since 11 is 2 plus 9 and 9 was already reachable. The reachable set after naming 2 is the reachable set after naming 11 and then 2, the only gap left is 1, and it is the opponent who faces it.

So on this position the argument does not merely prove a winner; unwound, it names the move. That is not general. It names the move here because the opponent’s winning reply to 11 happens to be unique and visible. On a position where the reply to F has to be searched for, the argument still proves the mover wins and still names nothing.

The pseudo-symmetric semigroup of 4, 5 and 7 goes the other way. Its Frobenius number is 6, and naming 6 wins outright: what is left has gaps 1, 2 and 3 — the semigroup generated by 4, 5, 6 and 7, which the census by genus found to be the only lost position of genus three. There the argument’s thought experiment is also the winning move, which happens on 123 positions of 583.

That is the stealing argument’s signature, repeated in a third game. It uses F as a thought experiment — if naming F loses, the reply that beats it was available first — and never claims F is good. On most positions it is not.

The top-right square, in arithmetic #

The correspondence with Chomp is close enough to be worth spelling out, because it says what kind of object the argument really needs.

In Chomp the stealing move is the top-right square. Every other move eats it — a move takes a square and everything above and to the right, and the top-right square is above and to the right of everything — so any position reachable by some move is also reachable by that move followed by another. Where the needle has a sentence found the needle for square bars and two-row bars and nowhere else, which is the usual distance between the argument and a move.

In Sylver Coinage, on an irreducible position, the Frobenius number plays the top-right square. Every move closes it, so any position reachable by some move is also reachable by naming F and then that move. The two games have nothing in common at the level of play — one eats chocolate, the other names integers — and the same structural fact about moves settles both.

The difference is where the structure comes from. Chomp has its dominated move at every rectangle because of the geometry of a rectangle. Sylver Coinage has one only on the irreducible positions, because of an arithmetic coincidence in the gap set. A reader handed a random Sylver position has no such move available, and the census says what that costs: one position in eight is lost.

A certificate for the verdict, and none for the move #

There is a way in which this result is worth more than the stealing arguments for Hex and Chomp, and it is about what a reader can check.

Deciding a Sylver Coinage position by search means recursing through every position its moves reach, each one recomputing which numbers have become reachable. Deciding whether a position is symmetric means comparing each gap x with F − x — a single pass along a strip of F numbers. So for irreducible positions the verdict the player to move wins comes with a certificate a reader can check in seconds, and the certificate is shorter than the question: a list of pairs.

That is the object a strategy is not a certificate says games generally lack — but only half of it. The pairing certifies the verdict. It certifies nothing about which move wins, and the 583 winning moves in the census are scattered across 123 cases of F, 112 of F less the smallest named number, and 348 of something else. In the vocabulary of what solved means these positions are solved in the stealing sense by a certificate and in the strong sense by nothing.

It is also a reminder that Sylver Coinage is an impartial game with ordinary Grundy values, which where the impartial theory stops takes as the place everything is supposed to be easy. The values exist and the recursion computes them. What the theory does not supply is a way to read a value off the semigroup without the recursion, and the pairing is the first thing found here that reads anything off it at all — one bit, for one class.

Not a line but the end of a gradient #

Symmetry is an extreme property, and a property that sharp usually sits at the end of something softer. It does here.

The type of a numerical semigroup counts its pseudo-Frobenius numbers: the gaps x such that adding any nonzero member of the semigroup to x lands back in the semigroup. F is always one of them, and a semigroup has type 1 exactly when it is symmetric.

The table sorts every position by type and genus parity. At type 1 the only loss is the genus-one position. Past it the two parities part company.

At odd genus the share climbs, though not steadily: about a fifth at type 2, a sixth at type 3, back to a quarter at types 4 to 6, and a third at types 7 and 8, rising further where the rows get thin. At even genus nothing climbs at all. Past type 1 the share sits between two and seven per cent at every type from 2 to 10, and the one larger share, in the row that gathers every type from 12 up, rests on three positions out of thirty-three.

A pseudo-Frobenius number is a gap sitting on the edge of the semigroup, one step from being absorbed by anything. At odd genus, where the parity makes lost positions common, the fewer such gaps a position has, the closer it is to the situation where every move closes F — and the rarer lost positions become. At even genus the parity already makes them rare, and the type has little left to explain. So irreducibility is the end of a gradient at odd genus and a sharpening of something already true at even genus. Neither half of that is a theorem; the endpoint is, and a proved endpoint on a measured tendency is a useful thing to know about a game where most of what is known is a table.

What the parity was, seen again #

The earlier essay’s parity has a mechanism — naming F always drops the genus by exactly one — and a first exception at genus eight. It is worth setting the new result beside it, because both are statements about the move F.

The parity mechanism says F is always available as a one-step move down in genus, and a game in which it were the only move would alternate perfectly. The stealing argument says that on irreducible positions F is contained in every move, so the whole game from there can be imagined as F first, then something. Neither says F is good. Both are facts about what F does to the set of moves, and together they explain why the move nobody needs to play organises so much of the census.

They also agree numerically where they overlap. A symmetric position has genus (F + 1)/2; irreducible positions occur at both parities; and at every parity the census finds them won. So irreducibility is not a restatement of the parity. It cuts straight across it — the 120 irreducible positions at genus fifteen and the 145 at genus sixteen are won alike.

What the census cannot say #

Sixteen is the limit. The argument covers every genus, and it is a proof: every move from an irreducible position closes F, for reasons of arithmetic that do not depend on size, and stealing does the rest. What the census adds is the check that the implementation agrees with the argument — and that nothing else in the census is being mistaken for this class.

The gradient by type is not proved at all. It is a table of shares at genus sixteen and below, and the rows past type ten are too small to read. Whether lost positions keep thinning as the type falls, at every genus, is a claim this page makes about the range it measured and no further.

Nor does anything here touch the opening. The positions in the census all have two coprime numbers already named and finitely many gaps. The empty position, and the position after a single number is named, have infinitely many gaps; they are not semigroups of finite genus; and the game that is a number system records that the opening 16 is still unsolved. The irreducible positions are settled by stealing and the opening is not, because the opening has no largest gap to close.

The rule the game is played under #

Sylver Coinage is normal play once the rule about 1 is read correctly: naming 1 loses, so nobody names it while anything else is available, and the player left with 1 as the only gap is the player without a real move. The stealing argument depends on that reading twice. It needs F to be a genuine move, which fails exactly when F is 1. And it needs the opponent’s winning reply y to be a genuine move too, which it is, since a reply of 1 would lose.

Under the other convention — the player who names the last nameable number loses, 1 included — the base case inverts, the argument’s two uses of F change meaning, and nothing on this page carries over without being redone.

Still open: what the pairing buys beyond the verdict #

The strip for 3 and 7 shows a symmetric position won by naming 2 and lost by naming 11, and the stealing argument is silent about why 2. The census knows the winning moves of all 583 irreducible positions and classifies them only coarsely — F wins on 123, F less the smallest generator on 112 more, and on 348 the winner is something else.

Whether the winning move of a symmetric position has a description — some gap chosen by the pairing itself, the way a pairing strategy names its replies in Chomp and in a board one column wider — is the question this leaves. The pairing of gaps around F is a symmetry of the position, and symmetries are where named moves usually come from. Whether this one names anything is not known here.