Eight squares, and no hotter

The obstacle was the catalogue swept every position reachable by play on four Domineering rectangles up to eighteen squares and found none hotter than $3/2$. It closed on the bound:

The rung above is the ceiling. Three halves on four boards up to eighteen squares is a bound with an argument behind it … turning that into a statement means finding the position that attains it and asking what its shape is. Every board here reaches $3/2$ and none exceeds it, so the attaining positions are in hand at four sizes, and if they are the same shape at every size then the bound has a witness and probably a proof.

They are the same shapes, there are five of them, and the ceiling holds two sizes past where a board sweep can look.

The ceiling is a fact about regions #

The first thing the attaining positions say is where to look.

Four hundred and forty-eight positions across the four boards reach $3/2$, and 428 of them have exactly one hot component. The rest of the board is numbers — dominoes’ worth of leftovers, single squares, regions already settled — which contribute their values and no temperature at all. A board is at the ceiling because one region on it is.

That is the sum is the object doing exactly the work it is for: a board with several regions on it is several games at once, and a question about the board is a question about whichever of them the answer lives in.

So the ceiling was never a fact about a board and asking it of boards was asking it in the wrong currency. Asked of regions, it can be pushed past the sizes a board sweep can reach: an eighteen-square board has seventeen thousand reachable positions and a ten-square region is one shape among nine thousand, evaluated once.

The other thing the boards say is when. On every one of the four, the ceiling is first reached at move two — as soon as each player has placed a domino — and never before. An empty rectangle is not the hottest thing on its own board; two moves in, it is. The reason is the one the rung below found for the whole arc: an empty board is symmetric, and a symmetric position is one where moving first buys little, because whatever one player does the other can mirror.

The hottest region of each size #

Take every Domineering region up to ten squares — 12,871 shapes — and record the hottest at each size:

The sequence is non-decreasing, which it has to be — a region contains smaller ones — and it rises by a quarter every two sizes up to eight. Then it stops.

Eight squares is where $3/2$ is first reached, and no region of seven squares or fewer is hotter than $5/4$. So the rung below’s bound is precisely the eight-square ceiling, met by the four boards because each of them can be carved down to one of these regions and no further.

What the sequence looked like it was doing #

A table ending at eight squares suggests a rule, and the rule is a good one: up a quarter every two sizes, starting from $1$ at four. It reproduces $1, 1, \tfrac54, \tfrac54, \tfrac32$ exactly and predicts $\tfrac74$ at ten.

Nine squares is not a test of it — the rule predicts $\tfrac32$ there too, and $\tfrac32$ is what happens. Ten is the test, and the rule fails it. Nine thousand two hundred and eighty-seven ten-square regions and the hottest is still $\tfrac32$.

That is what turns a bound into a finding. Three flat sizes running is not a sweep that ran out of room; it is a sweep that went one size past the point where a natural extrapolation said something else, and found the ceiling still there.

The five shapes #

Five regions of eight squares are at exactly $3/2$, counted up to the eight symmetries of the square. Each of them is a $2 \times 3$ block or a $4$-long bar with a domino’s worth of cells attached, and none of them is a rectangle: the hottest thing in Domineering is a lopsided region rather than a tidy one.

That is not a coincidence. A rectangle is a position both players can read — the values of every small board is the census of what they are worth — the whole of the anchor two rungs below is that a board’s temperature is about imbalance rather than about size. A region shaped so that one player’s placements interfere with the other’s, and one of them has slightly more room than the other, is a region where moving first matters — and moving first mattering is exactly what a temperature measures. How hot a day gets is the same question asked of the theory rather than of a game, where the hottest value born on a day rises without bound — so a game having a ceiling at all is the fact worth explaining.

Two of the five are never produced #

Three of the five shapes are the hot core of attaining positions on every board from the $3 \times 4$ to the $3 \times 6$. The other two appear on none of them.

They are at the ceiling and no sequence of legal moves on any of these rectangles reaches them. A domino removes two adjacent squares, and there are shapes a rectangle simply cannot be reduced to by removing dominoes from it — a region can be a legal Domineering position without being a reachable one.

This is a gap the site keeps finding in a different currency. The values nobody’s game produces measures it on values: the theory hands down 1,474 values born by day three and the rulesets reach under a tenth of them. Here it is on shapes, and the ratio is much less extreme — three fifths of the hottest regions are produced — but the phenomenon is the same one. What a subject allows and what a game makes are different sets, and the difference is where a census and a board part company.

When two pieces do it together #

Twenty of the 448 attaining positions have two hot components rather than one, and every one of them is on the largest board.

That is worth a paragraph because it is the one place where the ceiling is not a fact about a region. A board’s temperature is the largest of its components’ when one component is hottest; two components each below $3/2$ can combine to a board at $3/2$, because temperatures do not add and a sum’s temperature is not the maximum of its parts’ in general.

So the honest statement of the witness is: a Domineering position reaches $3/2$ when one of its regions is one of the five, or when two of its regions conspire. The first is the whole story on three of the four boards and 94 per cent of it on the fourth, and the second is a reminder that a decomposition is a tool and not a definition.

Why eight and not six #

There is an arithmetic reading of where the steps fall, and it is worth setting out because it is the closest thing here to an argument.

A Domineering region’s temperature measures what moving first is worth, and moving first is worth the difference between what the mover takes and what the opponent takes back. A domino covers two squares, so a single move changes the count of available placements by a bounded amount — and the bound is set by how many of the opponent’s placements the domino destroys.

A vertical domino in a region can destroy at most three horizontal placements: the two it lands on and the ones straddling either of its squares. So one move is worth at most about three placements’ worth of swing, and a temperature is half a swing. That is the shape of the rung below’s a move covers two squares, so what one move can be worth is bounded — and it does not give $3/2$, it gives something larger.

What the measurement adds is that the bound is not tight, and by how much. To realise a large swing a region has to hold a placement that destroys three of the opponent’s and be shaped so that the opponent cannot answer in kind. Eight squares is the smallest region with room for both conditions at once: fewer, and either the destroying placement or the asymmetry has to go.

That is a reading rather than a proof, and it is the kind this site is careful to label. What makes it worth writing down is the prediction it fails to make: it says nothing about why the ceiling stops at $3/2$ rather than continuing to rise as regions get larger and asymmetries get sharper. One fight makes a board a fight has the beginning of an answer — a large region breaks up rather than staying one fight — and turning that into a bound is the ladder’s remaining work.

Why a ceiling is a different claim from a maximum #

The result has two halves and only one of them is what a sweep normally establishes. Separating them says what is proved and what is merely very well supported.

The maximum over a sweep is a measurement. Every position examined has temperature at most three halves, one attains it, and the attaining regions are exhibited. That is as solid as an exhaustive sweep gets, and it is a statement about the positions swept.

A ceiling is a claim about every position of the game, at every size. The sweep supports it by the additional observation that the maximum does not move as the boards grow — it holds at eight squares, at nine and at ten, where a linear extrapolation from the smaller boards predicted seven quarters. A quantity that stops climbing where a naive model says it should keep climbing is evidence of a bound rather than of a slow increase.

But it remains evidence. Nothing here forbids a region of fourteen squares from being hotter, and the mechanism that would rule it out has not been supplied.

What that mechanism has to look like is worth stating, because it is unusually specific here. A temperature of $t$ needs a fight worth $2t$ to whoever takes it, so a hotter position needs a placement whose consequences are worth more than any placement on an eight-square region. The candidate argument is that a domino is a bounded object and a region larger than the reach of one placement supplies two independent fights rather than one bigger one — which would cap the temperature and would also explain why the attaining shapes are small and recur as cores of larger boards.

What a proof would need #

The rung below hoped the witness would bring a proof, and it is worth saying exactly how far it gets.

What a witness establishes is that the bound is attained — $3/2$ is not an artefact of a small sweep, it is a value a region actually has, and the shapes that have it are in hand. That is the easy half.

The hard half is that nothing exceeds it, and a witness says nothing about that. What would be needed is an argument bounding a region’s temperature by something about its shape, and the sequence here gives the shape of such an argument rather than the argument: the hottest temperature is non-decreasing in the region’s size, it rises in steps of a quarter, and it has now stopped for three sizes. Either it resumes — in which case the ceiling is a local plateau and the whole reading is wrong — or it is genuinely bounded, and then there is a quantity that bounds it.

The one hint the measurement offers is that the attaining regions are all eight squares. If the hottest region of every size from eight upward were an eight-square region with cold material attached, the ceiling would follow from a decomposition argument rather than from a bound on temperature. That is checkable at eleven and twelve squares and it is what the rung above should check.

What this does not say #

Ten squares is not large. A 3 × 6 board has eighteen squares, and the regions this page reaches are barely more than half that. The extrapolation refuted at ten is one extrapolation; a different one, rising every three or four sizes, is untouched.

A region is not a board. Every temperature here is of a region taken alone, which is what the decomposition licenses and what a player facing a broken-up board experiences. The four boards are swept whole, so the board numbers and the region numbers are the same measurement asked twice — but nothing here evaluates a twenty-square board.

The two unreachable shapes are unreachable from these four rectangles. A different starting rectangle, or a board that was not a rectangle, might produce them. What is measured is these boards do not, not no board does.

And no figure here can show a temperature. A region’s temperature is the height at which its thermograph’s walls meet, which is a fact about a diagram and not about the drawing of the region — so the pictures are the shapes and the numbers are beside them. How hot a real position is is where the diagrams themselves are drawn, and it is the page to read if the number wants a picture. Below zero is where the convention that a number has no temperature comes from, and it is the reason every maximum here excludes them rather than counting them as cold.

The convention, named #

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

A region is a four-connected set of free squares. A position’s regions are independent games and the position is their sum, which is what makes a temperature of a region a meaningful thing to quote.

A position’s temperature is the height at which its thermograph’s two walls meet. A number has none, and is excluded from every maximum here rather than counted as zero.

A region is at the ceiling when its temperature is $3/2$. A board position attains the ceiling when the whole position’s temperature is $3/2$, which happens when one of its regions is at the ceiling or when two hot regions combine to it.

Shapes are counted up to the eight symmetries of the square — four rotations and four reflections. The catalogue that enumerates them normalises by translation only, so it counts a shape and its mirror as two; the counts here say which convention is in use, and the symmetry count is the smaller.

Reachable by play means some sequence of legal placements produces the position from the empty rectangle, which is fewer positions than the occupancies: a set of covered squares that is not a union of dominoes never arises.

Where the ladder goes next #

The cold anchor has seven rungs: below zero, how hot a day gets, how hot a real position is, what a game actually produces, one fight makes a board a fight, the obstacle that was the catalogue, and now what the ceiling is a fact about.

The rung above is whether the hottest region of every size is an eight-square one with cold material attached. If it is, the ceiling follows from the decomposition rather than needing a bound of its own, and the argument is short. The test is the eleven- and twelve-square regions: enumerate them, find the hottest, and ask whether its hot part is one of the five. Twelve squares is 63,600 shapes, which is a sweep of a different order from this one and is affordable if the hot part can be found without evaluating everything.

Two neighbours are worth the trip. One fight makes a board a fight is where a board’s temperature was shown to be its hottest component’s, and it is what licenses asking the ceiling of regions at all. And what a game actually produces is the other measurement of the gap between a census and a board, and reading it beside the two unreachable shapes here is the clearest statement of what a played game leaves out.