The pairing removes moves it cannot name

Every move closes the largest gap proves that a Sylver Coinage position whose gaps pair off around the largest of them is a win for whoever is to move, and ends by admitting what the proof does not deliver. The winning move is unaccounted for. Naming F, the largest gap, wins on 123 of the 583 paired positions in range; F less the smallest named number wins on 112 more; on the remaining 348 the winning move is something the pairing says nothing about.

The obvious next question is whether some other reading of the same pairing names the move. The answer measured below is no, and it is a firm no — nine arithmetic rules, none reaching a quarter of the positions. What the pairing turns out to do instead is the opposite operation, and it is worth more: it does not say which move to make, it says which moves not to.

A result about positions, read as a result about moves #

The proved statement is about a position. Restated with the mover’s opponent in view it becomes a statement about a move.

Suppose a player names some gap x and the position this leaves has its own gaps paired around its own largest gap, which is above 1. Then the proved statement applies to that position, and the player it applies to is the opponent, who is now to move. So naming x hands the opponent a position the opponent wins, and naming x loses.

That is a test on a move, and it costs what the position test costs: one pass along a strip of numbers, comparing each gap x with F − x. It needs no search, it needs no recursion, and it is available at every position in the game rather than only at the paired ones — the position being played from can be any shape at all, because the test is applied to what the move reaches.

The position drawn above is the one generated by 5, 7, 9 and 11. Its gaps are 1, 2, 3, 4, 6, 8 and 13; there are seven of them, which is exactly (13 + 1)/2, so they pair off around 13 and the position is a win. Six of the gaps are legal moves. Naming 2, 3, 4 or 6 reaches a paired position, so all four lose without anything being searched. Two moves survive the test, and one of them wins.

Six candidates to two is the whole of what the test does here, and it is already a real reduction on a position where the proof itself was silent. The question is what it does everywhere.

One move in six #

Over the whole census the test removes 27,215 of 159,728 moves, which is 17.0 per cent of every move anybody could make from any position in range. Not one of the 21,234 winning moves is among them.

That last count is the point, and it is a count rather than a reassurance. The soundness of the test is an argument — a struck move reaches a position already proved to be a win for whoever moves next — and an argument that had a hole in it would show up here as a winning move with a line through it. There is none, at any of the sixteen sizes.

The share falls steadily as the positions grow: five moves of eight at three unnameable numbers, two in five at eight, a third at ten, a seventh at sixteen. Paired positions get rarer as the genus rises, so the moves that land on one get rarer too. That decline is the honest limit on the whole idea, and it is visible in the column rather than left to be worried about.

The argument’s own move, struck off by the argument’s own test #

The position every move closes the largest gap opens with is the one generated by 3 and 7, and it shows the test doing something the proof would not lead anybody to expect.

The gaps are 1, 2, 4, 5, 8 and 11, and the proof about this position is conducted entirely through the move 11. If naming 11 loses, the reply that beats it was available first — that is the whole argument, and it wins the position.

The test strikes 11 off. Naming 11 leaves gaps 1, 2, 4, 5 and 8, five of them, with largest gap 8; and 5 = (8 + 2)/2, so those gaps pair off around 8 with 4 as the odd one out. The position is pseudo-symmetric, the opponent wins it, and so naming 11 loses.

Both statements are true and they are about different things. The proof uses 11 as a thought experiment and never claims it is good; the test evaluates 11 as a move and says it is bad. The proof and the test are two readings of one property, pointing in opposite directions, and a reader who took the proof to be advice about play would be taking the losing move on this position every time.

That is the general shape of a stealing argument, seen from a new side. The theorem that names no move and a board one column wider both end at the same place — a winner named, a move not — and neither asks what the argument’s own move is worth when actually played. Here it is worth a loss, on 460 of the 583 positions.

What the descriptions are worth #

If the pairing will not name the move, perhaps something else read off the semigroup will. Nine rules were scored, each naming a gap, or a handful of gaps, from the semigroup alone and with no search allowed.

The largest gap wins on 123. The largest gap less the smallest named number wins on 112. The second-largest gap, which sounds like a different idea and is often the same number, wins on 114. Half of F rounded down wins on 67 of the 394 positions where it is a legal move at all. The smallest gap above 1 wins on 35, and the flat rule name 2 wins on 20.

Two rules do better and both do it by naming several gaps rather than one. A gap that becomes nameable when any named number is added to it — a pseudo-Frobenius number, the quantity whose count is a semigroup’s type — catches a winner on 154. F less a minimal generator of the semigroup catches one on 279, which is 48 per cent, out of a list averaging four candidates. A rule that offers four guesses and is right about half the time is not a description of a move; it is a shortlist, and a shortlist is what the test already produces more cheaply.

No rule naming one gap reaches a quarter of the positions. That is a measured negative over the whole range, not an admission that nobody thought hard enough, and the figure under it says what is standing in the way.

On 367 of 583 positions there is exactly one winning move. A position in this class offers 12.3 legal moves on average, so a rule guessing at random would be right about one time in twelve, and the best rule naming a single gap is right about one time in five. The difficulty is not that the winning move is hard to describe among several equally good ones. There is usually nothing else to choose.

And one line of that figure is sharper than the rest. Where naming the largest gap wins, it is the only move that wins — all 123 of 123. The move the stealing proof is built around is never one winner among several. It is either forced or useless, and the proof cannot tell a reader which, because telling them apart is exactly the search the proof was standing in for.

What is predictable is the size, not the name #

One thing about a winning move can be predicted without any search, and it is not which gap it is.

A parity with a first exception found the count of unnameable numbers very nearly deciding the game: positions with an odd count are thick with losses and positions with an even count are nearly empty of them. A winning move is a move to a lost position, so a winning move nearly always lands on an odd count — 775 of the 981 winning moves here do.

So the parity of what a winning move closes is nearly forced, and it is forced in opposite directions from the two kinds of position. From an odd count a winning move mostly closes an even number of gaps; from an even count, an odd number. That is a real constraint and it is worth exactly as much as it sounds: it halves the shortlist and says nothing about which half.

A property that predicts the size of a move without predicting the move is the recurring shape of this game. The count of unnameable numbers predicts the outcome without predicting the play; the pairing predicts the outcome without predicting the move; and the parity of the drop predicts the shape of the move without predicting the number. Three separate readings, each one bit short of useful, which is what makes where the impartial theory stops worth reading against a game that has ordinary values and no way to read one off a position.

Two positions lost with no search at all #

On two positions in the whole census the test removes every legal move.

If every move is struck, every move loses, so the position is lost for the player to move. The verdict follows from four one-pass tests and no recursion whatever. It happens on the position generated by 4, 5, 6 and 7, whose gaps are 1, 2 and 3, and on the one generated by 4, 5 and 11, whose gaps are 1, 2, 3, 6 and 7.

Two out of 11,769 is not a technique. It is worth recording for one reason: it is the only place in this game where the pairing certifies a loss. Everywhere else the pairing certifies wins — that is what the stealing proof does — and a stealing argument is structurally incapable of proving anybody loses, since it works by showing that whatever the opponent could do, the mover could have done first. Applied to the move list rather than the position, the same property proves the opposite kind of statement twice, and the two positions it proves it on are small enough to check by hand.

What it saves a search that still has to run #

The test does not replace the search. It removes moves from it, which is a different and more ordinary kind of help — the same kind how much a list of options can lose prices for dominated options and the order a solver tries the moves in prices for ordering.

95,516 descents become 72,005, a saving of 24.6 per cent, and the two solvers agree on every position in the census. The saving is larger than the 17 per cent of moves removed, because a struck move is never descended into and so its whole subtree is never reached either.

It is worth being precise about what kind of saving that is. A struck move is a dominated move — known bad before it is examined — and removing it is sound for the full search, including for proving a position lost, because a bad move need not be tried when every move has to be shown bad. It is not a heuristic ordering and it is not a cut-off; nothing is approximated, and the two solvers return the same verdict on all 11,769 positions rather than agreeing to within something.

Against that, one pass a move is not free. The pruned solver performs 95,516 pairing tests — one per move considered, the same number as the plain solver’s descents — so the trade is a recursion for a scan. In a game where a recursion re-derives the whole reachable set below a position and a scan walks a strip of at most thirty numbers, that trade is worth taking; in a game where the two cost the same it would not be.

What the measurement cannot say #

Sixteen unnameable numbers is the limit, and the falling share matters. The soundness of the test is a proof and holds at every size. Its usefulness is a measurement, and the measurement is declining: three moves in five removed at the smallest positions, one in seven at the largest. Whether it keeps falling, and how fast, is not settled by a range that stops where this one does.

The nine rules are nine rules. A negative result about a list of descriptions is a statement about that list. It is evidence that the winning move has no simple arithmetic description and it is not a proof of one, and the two rules that name several gaps are the reminder that the boundary between a description and a shortlist is where such a result gets soft.

And the opening is untouched. Every position here has two coprime numbers already named and finitely many gaps. The game that is a number system records that the opening 16 has been worth a thousand dollars since 2017, and nothing on this page reaches it: a position with infinitely many gaps has no largest gap, so it has nothing to pair and no test to apply.

The convention the verdicts are computed under #

Normal play, with the rule about 1 read the way the game is actually played: naming 1 loses at once, so nobody names it while any other number remains, and the player left with 1 as their only option is the player with no real move. Every count on this page — winning moves, struck moves, descents — excludes 1 from the move list for that reason.

The test itself depends on the convention twice. It needs the position a move reaches to be a win for whoever moves next, which is the stealing proof’s conclusion and is stated under normal play. And it needs the exception, which is why the test asks for the largest gap of the position reached to be above 1: the position whose only gap is 1 is the one paired position that is lost, and a move reaching it is a winning move rather than a struck one. Seventy moves in the census reach it, and the exception is what keeps every one of them out of the struck column.

The surprise: the same property, used backwards, is worth more #

The stealing proof is the strongest result anybody has about this game’s finite positions, and it is a result of a kind that is famously unusable — a strategy is not a certificate measures the gap between knowing a winner and having a strategy, and three different claims are all called solved separates the two as a matter of definition. A verdict with no move is the weakest useful thing a theorem can produce.

Read backwards it produces something the forward reading does not. The forward reading applies at 583 positions, tells the mover they win, and leaves a dozen moves to search. The backward reading applies at every position in the game, tells the mover nothing about the outcome, and removes a sixth of the move list for a scan — including, on the two smallest cases, all of it.

That is a trade worth naming, because it is available whenever a theorem settles a class of positions. A theorem that says who wins a class of positions is also a theorem that says which moves not to play, everywhere, since a move into that class is a move whose verdict is already known. The forward form is the one that gets stated because it is the one that sounds like a result. The backward form is the one that gets used, it applies far more widely, and here it is measurably the more valuable of the two — a quarter of a search, against a proof that names no move at all.

Still open: whether the shortlist can be closed #

The test leaves a shortlist and the parity of the drop halves it. On the position generated by 5, 7, 9 and 11 that shortlist has two moves on it; averaged over all 583 it has 9.6, against 12.3 legal moves. Nothing here decides whether some further one-pass property closes the list to one.

The measurement that would begin to settle it is a census of the shortlists themselves: for every paired position, the moves that survive the test, sorted by whether the survivors have anything in common beyond surviving. If the winning move is always the survivor that closes the fewest gaps, or always the largest survivor, that is a rule and it is one scan away. If the survivors look like the gaps did — a dozen numbers with one arbitrary winner among them — then the shortlist is as far as a scan reaches, and the remaining work is the search this game has never been able to avoid.