The obstacle was the catalogue

One fight makes a board a fight measured the temperature of played Domineering boards by evaluating each of their components against a catalogue of small regions, and could say nothing about half its positions, because their components were too large for the catalogue to hold. It closed on those:

The rung above is the early game, and it needs a bigger evaluator rather than a better idea. Half the positions here are unsettled, they are all early, and the question they hold is whether a board that has not yet broken up is hot … A region of twelve squares is out of reach of the exact evaluator and within reach of a bounded one, so the honest next step is a bracket rather than a value.

No bracket is needed, and there was never anything wrong with the evaluator.

What was actually expensive #

A twelve-square Domineering region evaluates in about five milliseconds. An eighteen-square one takes under a second. Neither is out of reach of the exact evaluator and both were out of reach of the catalogue, which is a different object entirely.

There are 369 polyominoes of eight squares, 4,655 of ten, and 63,600 of twelve. A catalogue must hold every shape a board might break into, so its cost is the count of shapes; a sweep over positions holds only the shapes a board actually reaches, and pays once for each. The rung below’s obstacle was that its method scaled with the shapes rather than with the game.

And the two costs are not merely different in size. A catalogue of shapes shares nothing between its entries — each shape is evaluated from scratch. A game tree evaluated once memoises every subposition every other position needs, so sweeping all 17,014 positions of a three-by-six board costs barely more than evaluating the empty board alone. Four boards up to eighteen squares, 26,318 positions in total, take under two seconds.

The board cools as it is played #

Sweeping every reachable position and evaluating the whole board at each one gives the arc a player would describe.

On a three-by-six board the hot share is 67 per cent after one move, 79 after two, 79 after three, 63 after four, 39 after five, 14 after six, 2 after seven, and nought thereafter. The opening is a fight and the endgame is a number, and the change happens over about three moves in the middle.

The same shape appears on all four boards, with the peak arriving later on the larger ones — move one on a 3 × 4, move two on a 3 × 5, move three on a 4 × 4 and a 3 × 6. That is what a phase of the game looks like rather than an accident of one board, and it puts a number on something how hot a real position is could only describe: the hot positions are early, and early means the first two or three moves.

And breaking up is what cools it #

The rung below’s question was specific: is a board that has not yet broken up hot? It is, and being in one piece is not merely correlated with being hot — it accounts for much of the difference.

On the three-by-six board a position still in one piece is hot 47 per cent of the time and a broken one 36 per cent, across every position of the game. The gap holds on all four boards and widens with the board — 33 against 18 on the smallest, 47 against 36 on the largest.

The mechanism is temperatures do not add read backwards. A sum’s temperature is the largest of its parts’, so breaking one hot region into two cannot make a board hotter than its hottest piece — and the pieces are smaller than what they came from, so their temperatures are smaller too. Splitting a region is the operation that converts one large fight into two smaller ones, and two smaller fights are worth less to move in than the one they replace.

It is worth being careful about what that does not say. Being whole is not sufficient: on the largest board fewer than half the whole positions are hot, because a whole position late in the game is a long thin corridor with nothing at stake in it. What the count establishes is that connectivity is a real term in the answer and one a player can read off the drawing without evaluating anything.

The empty board is the cold one #

The arc has an exception at its very start, and it is a good one. The empty 3 × 4 is worth $-3/2$, the empty 3 × 5 is worth $-1$, and the empty 4 × 4 is a long form with a temperature of exactly nought. Only the 3 × 6 is hot. Yet one move later, two thirds of the positions on every one of those boards are fights.

The reason is symmetry. An empty rectangle is the most symmetric position the game has, and a symmetric position is one where neither player gains by moving first — the strategy that is a symmetry is where that argument is made in its strongest form. Heat arrives with the first move, which breaks the symmetry and creates the imbalance a temperature measures, and leaves again when the board breaks into pieces too small to be worth fighting over.

So the honest shape of a Domineering game is not hot, then cold. It is cold, hot, cold, and the cold opening lasts exactly one move.

Two ways to answer a question about a sum #

The two rungs use opposite methods on the same game, and it is worth setting them side by side because neither is simply better.

The catalogue method evaluates each shape once, stores it, and answers any board by decomposing it and looking its pieces up. Its cost is the number of shapes, which triples with each square; its benefit is that once built it answers boards of any size, because a 10 × 10 board late in a game is made of the same small pieces as a 5 × 5. That is exactly what the board falls apart is for, and it is why the rung below could measure a 6 × 6 at all.

The sweep method evaluates positions and never decomposes anything. Its cost is the number of reachable positions, which grows with the board far faster than a catalogue does; its benefit is that it needs no pieces to be small. It cannot reach a 6 × 6 board — that game tree is far beyond an exact evaluator — and it answers the early game on an 18-square board completely, which the catalogue cannot do at all.

So the two methods have complementary blind spots, and the rung below hit the catalogue’s. The right reading is not that the catalogue was a mistake but that the question changed: a question about late positions on large boards wants a catalogue, and a question about early positions wants a sweep. Asking the second question with the first instrument is what produced the impression that a bigger evaluator was needed.

There is a general form of that worth keeping. A method’s limit is not usually the limit of the subject, and the way to tell them apart is to price the obvious alternative before concluding that a question is out of reach. Five milliseconds against 63,600 shapes is not a close call, and nobody had put the two numbers next to each other.

The same ceiling everywhere #

Across 26,318 positions on four boards, not one is hotter than $3/2$. The board grows by half from the first to the last and the ceiling does not move at all.

That is a strong statement about what a Domineering temperature measures. A quantity that grew with the board would be measuring how much of the board is left; this one measures what a single move is worth, and a move covers two squares whatever the board is. So the temperature is bounded by what one domino can be worth to place, and on these boards that turns out to be a move and a half.

Whether $3/2$ survives to larger boards is the obvious question and is not answered here — four boards is four points, and the largest of them is eighteen squares. What can be said is that the ceiling did not move over a fifty per cent increase in board size, which is the kind of stability a bound has and a coincidence usually does not.

What a player takes from the arc #

The measurement is a census and the reason to want it is practical, so it is worth saying what a player would do with it.

Three things follow from hot early, cold late. The first is that the opening is where calculation pays: a hot position is one where moving first is worth something, so the difference between the best move and a plausible one is largest there. The second is that the endgame is where a count replaces a calculation — once every region is a number the board is the sum of those numbers, and numbers avoid numbers says neither player wants to move at all, so the remaining play is bookkeeping. The third is that the transition is fast: on a three-by-six board it takes three moves to go from four fifths hot to a seventh.

That third one is the surprise, and it is the reason the arc is worth a figure rather than a sentence. A player might reasonably expect the heat to drain away gradually as the board fills, in proportion to the squares left. It does not: the hot share is still 63 per cent with four dominoes down and two thirds of the squares covered, and 14 per cent two moves later. What changes in those two moves is not how much board is left but whether it is still in one piece.

Cataloguing and sweeping are different questions #

The finding here is that one measurement was expensive and a different one is cheap, and it is worth stating the distinction generally, because the two look alike and this ladder confused them for three rungs.

Cataloguing asks: what is every shape of a given size worth? The cost is the number of shapes, which grows like the shapes do — a factor of three and a half a square for polyominoes — so a catalogue to twelve squares is out of reach whatever each evaluation costs.

Sweeping asks: what are the positions a board actually reaches worth? The cost is the number of positions reached, which is bounded by the play-outs rather than by the combinatorics, and a play-out visits a few dozen positions however large the board is.

The two agree only if a game reaches most of its shapes, and no game does. A Domineering board reaches a small, structured, repetitive set of regions, so the sweep is cheap for the same reason the catalogue is dear: the shapes vastly outnumber the ones that turn up.

That is why the early game is unmeasurable was wrong rather than merely pessimistic. The early game’s regions are large, so they are far out in the catalogue and the catalogue could not reach them; there are very few of them, so a sweep reaches all of them in seconds.

The instruction is to ask what a measurement is a measurement over, before deciding it is too expensive. A cost that grows with the space of possibilities is not the same as a cost that grows with what is visited, and in this subject the gap between the two is the whole reason exact evaluation is ever affordable.

What this does not say #

These boards are small. Eighteen squares is a third of a real 8 × 8 Domineering board, and every arc measured here fits into nine moves. A larger board would have a longer hot phase and might have a different shape in the middle; what it cannot have, if the ceiling holds, is a hotter one.

Every position, not every game. The sweep counts positions reachable by some sequence of moves, weighting each equally. A played game visits one path, and a path chosen by good play is not a uniform sample — one fight makes a board a fight measured the played population and found a lower hot share for exactly that reason. The two measurements answer different questions and both are wanted.

The connectivity finding is a correlation with a mechanism, not a cause. Whole positions are also, on average, earlier positions, and earliness is itself a strong predictor. The gap survives being read board by board, and it has not been read at a fixed move number on every board — at move two on the smallest board the few broken positions are actually the hotter ones, which is why the claim is stated over the whole game rather than move by move.

The empty-board exception rests on three cases. Three of four empty rectangles being cold is a small sample, and the symmetry argument it is read through is a sketch rather than a proof — an empty rectangle is symmetric under a reflection that does not swap the two players, so it is not the classical symmetry argument at all. What is solid is the measurement: the empty board is cold on three of the four and the boards one move later are fights.

And the whole board is evaluated rather than added. Nothing here uses the decomposition the rung below relied on, so nothing here is a check on it. A board’s temperature computed whole and computed as the maximum over its components should agree, since temperatures take a maximum over a sum — and that agreement is assumed rather than measured.

The convention, named #

Normal play throughout: a player who cannot place loses. Left places vertical dominoes and Right horizontal ones.

A position is a board with some squares covered, and the sweep is every position reachable by any sequence of legal placements from the empty board — not every occupancy, since a set of covered squares that is not a union of dominoes never arises.

The move number is how many dominoes are down, so it counts both players’ moves together.

A position is whole when its free squares form a single connected piece, four-connected, counting single squares as pieces of their own; broken otherwise. Both are read off the drawing.

Hot means the temperature is above nought, which for a Domineering board is the same as the position not being a number — somebody wants to move in it. A number and a position of temperature nought are both counted as cold, which is the convention how hot a real position is uses and is why the empty 4 × 4 counts as cold despite not being a number.

Where the ladder goes next #

The cold anchor reaches six rungs to here, and this one has just turned the whole heat picture round by measuring the early game. The rung above supplies the ceiling.

Eight squares and no hotter takes the observation that no Domineering position anywhere in this ladder’s sweeps exceeds a temperature of three halves and asks which position attains it. It is a region of eight squares, there are five such regions up to symmetry, and three of them are the hot core of an attaining board at every size swept — so the maximum is realised by a small local shape rather than by a large board arranging itself well.

The part that could not have been guessed is that the ceiling holds at nine and ten squares, where the obvious extrapolation from the smaller boards predicted seven quarters. So the temperature of this game does not climb with room: past eight squares the extra space adds positions and adds no heat, which is the opposite of what the construction’s own record does, where a day buys a degree without limit.

That contrast is the anchor’s closing statement. A day’s hottest value is about how far the integers get; a game’s hottest position is about how much board a fight can occupy — and the second has a limit that the first does not, because a fight needs its two sides to be genuinely available and a board large enough to hold two big fights holds two, not one bigger one.