The winning reply is the fourth choice

A potential that names every move hands Down a rule that needs no symmetry and no table: weigh each of Across’s unfinished chains by two to the minus the number of cells it still needs, and take the empty cell whose live chains weigh most. The guarantee behind it reaches only boards two rows deep. It keeps winning far past that — every three-row board Down can win at all — and then loses a four-by-five board that a table of pairs holds with certainty, by answering in the middle and giving up the bottom edge.

It ends by naming the repair that suggests itself. A chain along an edge has fewer ways to be blocked than one through the middle, so weigh it more. Fifty-five weightings later the board is still lost, and the reason turns out to be somewhere else entirely.

The bonus, and what it costs #

A chain’s weight here is a product, one number for each cell Across has not yet claimed, so raising the cells on the outer rows raises exactly the chains that run along them. At every cell one half it is the potential unchanged, which is the control and is checked against the plain rule’s own verdicts on the same six boards.

No bonus holds four by five. At a quarter it loses, at a half it loses, at one and at two it loses.

And it is worse than a failure to help. At a bonus of one half the potential stops holding the three-by-four, three-by-five and three-by-six boards, which it held at no bonus, and at a bonus of two it stops holding the two-by-three as well. Four boards lost to buy a board that was never bought. The repair is not neutral where it fails; it is actively harmful, because weighing the edges more means neglecting the middle, and the middle is where the three-row boards are decided.

The last column of the table is the same rule with the weights removed altogether — take the cell that lies on the most live chains, counting rather than weighing. That holds one board of the six. The weights are doing real work; they are simply not doing the work the repair needed.

Every weighting in reach #

An edge bonus is one family and the question deserves the wider one. Give every cell its own number, let a chain weigh the product of those numbers over the cells Across has not claimed, and let Down take the heaviest empty cell: that is the potential’s rule with the half replaced by anything at all.

Fifty-five such weightings were run, each against every line Across can play. None holds the board. Varying the outer rows against the inner ones does not do it at any of twenty-five settings; varying the outer columns against the inner ones does not do it at any of twenty-five; and neither does changing the half to a third, to two fifths, to three fifths or to seven tenths.

That is not a proof that no weighting works — the family is a grid and the grid is finite. It is enough to stop looking in that direction, and the next measurement says why nothing in that direction was ever going to be found.

What the winning strategy is actually asking for #

The pairing that holds four by five is exact: Across plays a cell, Down plays its partner, and Across never joins the two sides. So there is a winning strategy to compare the potential with, move for move, and the comparison is the whole diagnosis.

At 393,660 of Down’s turns the potential would choose the pairing’s cell 102,768 times. Down faces seven empty cells at the average turn, so a rule taking one at random would be right 14.3 per cent of the time. The potential manages 26.1, which is most of a factor of two better and is not a strategy.

The other column of that table is the one that settles it. The winning cell sits 3.68th in the potential’s order on average, and taking a cell at random would put it 4.00th. Judged by where it ranks the move that actually holds the board, the potential’s whole ordering of the position is worth about a third of a place against chance. It is right at the top more often than chance and wrong further down by about as much, and the two nearly cancel.

A re-weighting moves cells up and down that order by a little. It does not move the fourth to the first while leaving the rest of the board’s play intact, and it certainly does not do so at the turns where the winning cell is twelfth or seventeenth.

The shape of the distribution says the same thing again. The counts do not fall away smoothly from the top: rank 1 takes 102,768 turns, rank 2 only 48,654, and rank 3 climbs back to 73,311; rank 4 drops to 35,735 and rank 5 rises again to 48,878. A rule whose errors were small perturbations of the right answer would give a decreasing column. This one alternates, which is what a comparison between two unrelated orderings looks like — and the alternation is a parity effect, because the pairing’s partner is a fixed cell and the potential’s ranking of it depends on how many of the chains through it Across has already half-built.

That is the answer to the question this page was set. The board is not lost to the numbers. It is lost to the rule that ranks cells by a number at all, and a rule of that shape cannot be tuned into this pairing because the pairing is not ordering cells by anything the potential measures. It is answering a cell with its partner, and the partner’s weight is an accident.

Taking the numbers away #

The last column of the first table deserves its own reading, because it is the control for everything above.

Strip the weights out and keep the rule: at each turn Down takes the empty cell lying on the most live chains, counting them rather than weighing them. That is the potential with every chain worth the same, and it is a rule somebody would write down first and refine afterwards.

It holds one board of the six — the two-by-five — and loses the two-by-three, which the potential holds at every bonus up to one, and every three-row board, which the potential holds at all of them. So the exponential weighting is worth four boards, and the discovery of this page is that those four boards are the whole of what it is worth: it buys a great deal against counting and nothing at all against the four-by-five.

That is the useful shape of the result. The potential sits between a rule with no numbers in it and a strategy with no numbers in it — the pairing — and it is much nearer the first. Adjusting its numbers moves it around inside that neighbourhood.

Why the potential is good anyway #

It would be easy to read all this as a verdict against the potential, and that is the wrong reading.

Erdős and Selfridge proved that a blocker moving second who plays this way never loses when the total weight at the start is below one half. On four by five that total is 1.992, four times the guarantee, over 148 minimal chains. The theorem says nothing whatever about such a board, and the potential holds eight boards past the line before it fails on the ninth.

The total is worth reading across the whole sweep, because it says where the potential is really operating. Two rows deep it falls as the board widens — 0.563 at two by three, 0.438 at two by four, 0.270 at two by six — so the guarantee bites on the wider boards and the potential is proved there. Three rows deep it sits just above one at every width: 1.078, 1.008, 0.945, which is close enough to the half to be another world. Four rows deep it is 1.906 and 1.992, four times the line, and 148 minimal chains rather than 25.

The three-row boards are where the potential is doing the work nobody proved it could do, and the four-row boards are where it stops. Nothing in the total predicts that: three by three has a total of 1.156 and Down loses it because Down cannot win it at all, while four by five has 1.992 and Down can. The total says how far past the guarantee a board is; it says nothing about whether the rule holds there, and the only way to find out is the walk over every line.

A rule that keeps working four times past its own guarantee and then fails is behaving exactly as a rule that is never right and cannot be far wrong describes: the bound is what can be proved, the performance is what is measured, and the gap between them is where most of the useful behaviour lives. How much a list of options can lose prices the same trade from the other side, where a rule throws options away rather than choosing among them. What this page adds is the shape of the far side of that gap. The potential does not degrade gracefully into the pairing’s territory; it does something structurally different, and the rank distribution is how different.

Two kinds of construction, and what each needs #

Three constructions have now named moves where strategy stealing names none, and they need different things.

A pairing needs a symmetry of the board, and where it exists it is exact. A board one column wider is the case in point: on an n by n + 1 board Down’s edges are nearer together and the pairing follows from that, checked against every line. It names every reply and it is never wrong. Where the needle has a sentence finds the same trade in Chomp: a pairing that names every move on the families with the right symmetry, and nothing at all on the rest.

A tree needs the game to be about connection in a particular way. A winning strategy that is a spanning tree is exact for the same reason and needs a graph with two edge-disjoint spanning trees in it.

A potential needs nothing at all. It is defined on any board, it names a move at every turn, and it is not exact. That is the trade the theorem that names a winner and no move sets up and it is a real one — but the measurement here shows the trade is sharper than sometimes wrong. Where the potential is wrong, it is wrong about the ordering of the whole position and not about one cell, because its choice and the winning choice agree only a quarter of the time even on plays where the potential is not yet lost.

What the search cannot say #

Fifty-five weightings is fifty-five weightings. They cover three natural families on a grid of five values, and a weighting outside those families — one that looked at how many chains a cell lies on, say, and weighted by that — is untested. What is established is that the proposed repair, and the obvious generalisations of it, do not work.

And the rank distribution is measured under the pairing’s play, not under the potential’s. It says what the winning strategy asks for at positions the pairing reaches; it does not say what the potential would need at the positions the potential itself reaches, which are different positions after the first divergence.

And the comparison is with one winning strategy. The pairing is a strategy that holds four by five; there may be others, and a cell the pairing declines might hold the board too. The rank distribution therefore overstates how wrong the potential is at any single turn, and the overstatement is bounded by how many winning replies a position has — which is not measured here. What it does not overstate is the divergence on the losing line, where the potential’s cell is checked and does lose.

Twenty cells is the ceiling. Every measurement here is on boards of at most twenty cells, because the walk over every line of Across’s is what makes a claim about a rule a claim about all of Across’s play. Five by six, where the pairing is fifteen pairs, is past that.

The convention the boards are played under #

Hex has no draws — a full board always contains a chain for one side or the other — so the game is decided and there is nothing between winning and losing. Across joins left to right, Down joins top to bottom, and on a rectangular board those are different tasks, which is the whole content of a board one column wider.

Down moves second everywhere here. That is what the Erdős–Selfridge theorem is about — it bounds a blocker who replies rather than one who opens — and it is what makes the four-by-five board interesting: a board where the second player wins is a board where strategy stealing says nothing, because the argument gives the win to the first.

The chains counted are the inclusion-minimal ones. A chain containing a smaller chain never matters, and counting both would weigh some cells twice.

The surprise: a good heuristic and a winning strategy need not resemble each other #

The intuition behind repairing a heuristic is that a rule getting most positions right is close to a rule getting all of them right. Move a weight, shift a threshold, break a tie the other way, and the remaining errors go.

That intuition requires the heuristic and the exact strategy to be ordering things similarly, and here they are not. The potential is a good rule — it holds eight boards past its guarantee — and the exact strategy on the ninth agrees with it a quarter of the time. Those two facts sit together uncomfortably and they are both measured.

The resolution is that the potential and the pairing are answering different questions. The potential asks which empty cell is doing Across the most good, and blocks it. The pairing asks which cell did Across just take, and answers its partner. On boards where blocking the most valuable cell happens to be answering the partner often enough, the potential wins; on four by five it does not, and no amount of adjusting what valuable means will make a weight function into a lookup by opponent’s move.

Three different claims are all called solved separates knowing a winner from having a strategy, and this is a third thing again: having a rule that wins, and having a rule that plays the winning moves. A heuristic can be near-optimal in outcome and unrelated in behaviour, and the distance between them is not visible in a win rate. Measuring it needs an exact strategy to compare against, which is why this measurement is available on four by five and nowhere larger.

Still open: whether anything ordered by a number can hold the board #

The negative here is about products of per-cell numbers and about the counting rule. It does not settle the wider question: is there any function of the position at all, computed cell by cell, whose maximum Down can take and hold four by five?

The measurement that would settle it is a search over functions rather than over weightings — for each position the pairing reaches, the set of cells that keep Down winning, and then whether any consistent ordering of cells puts one of those first at every such position. That is a question about whether a system of constraints has a solution rather than about a grid of parameters, it is finite on this board, and it would answer for every rule of the potential’s shape at once rather than for fifty-five of them.