The ceiling was a plateau

Eight squares, and no hotter enumerated every connected Domineering region of at most ten squares, found the hottest of each size to run $0$, $1$, $1$, $5/4$, $5/4$, $3/2$, $3/2$, $3/2$, and read the last three as a ceiling — a rise of a quarter every two sizes that stops at eight. It closed by naming the test:

The rung above is whether the hottest region of every size is an eight-square one with cold material attached … 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.

The eleven-square regions have been enumerated. There is no ceiling to explain.

Seven quarters at eleven #

The sweep runs over every connected region of at most eleven squares — 46,924 of them, 34,053 at the top size alone — with each region’s temperature taken from its own thermograph.

At eleven squares the hottest is $7/4$.

So the $3/2$ that held at eight, nine and ten squares is a plateau three sizes wide rather than a bound, and the sweep that found it had ended inside it. The sequence over sizes three to eleven is

$$0, \ 1, \ 1, \ \tfrac54, \ \tfrac54, \ \tfrac32, \ \tfrac32, \ \tfrac32, \ \tfrac74$$

and the rise resumes at exactly the size the rung below could not reach.

Four shapes attain it, in two mirror pairs, worth $\{3{\ast} \mid -1/2\}$ and its negative. That is a much smaller attaining set than the forty-three shapes attaining $3/2$ at ten squares or the eight at size eight — and the smallness is itself a signal. A temperature attained by four shapes in thirty-four thousand is a step that has just been taken; a temperature attained by forty-three is a plateau being sat on.

What a region’s temperature is #

The quantity is worth restating, because eight rungs of this anchor have been about it and it is easy to read as a size.

A region’s temperature is how much is at stake in it: the height at which the two walls of its thermograph meet, which is the ambient temperature above which neither player wants to move there. A cold region is one both players are content to leave alone; a hot one is a fight. It is not the region’s value, and it is not how large the region is — below zero is where the two are separated on this anchor.

What makes a maximum over all regions of a size interesting is that it bounds what a Domineering board can be worth fighting over. A board falls into regions, its temperature is its hottest region’s, and so a bound on regions of size $n$ is a bound on any board whose largest piece is $n$ squares. That is the chain the rung below was building and it still holds; what has changed is the number at the end of it.

Contained, and hotter #

The rung below’s picture was that a large hot region is a hot eight-square one with cold material attached, which would make the ceiling a consequence of the board falling apart rather than a bound needing its own argument. Half of that picture is exactly right.

Every one of the four eleven-square attainers holds five of the eight hottest eight-square shapes as a connected sub-shape. The containment is not a near thing and it is not one core each — the hot eight-square shapes overlap heavily, so a hot eleven-square region contains several of them at once.

And the region is a quarter of a move hotter than the shape inside it. So the extra three squares are not cold material; they are what makes the region hotter than its core.

That is where the explanation fails. A containment explains a ceiling only if the extra material changes nothing, and here it changes exactly the step size the sequence had been taking all along.

The step is a different kind of hot #

There is a reading of the four attainers that makes the break less surprising, and it is worth having because it suggests where the next one will be.

The eight-square attainers come in two kinds. Three of the five orbits are worth $\{3/2 \mid -3/2\}$ — symmetric fights, where the two players’ prospects mirror each other. Two are worth $\{0 \mid -3\}$ or $\{3 \mid 0\}$ — lopsided ones, where one player has much more to gain than the other.

All four eleven-square attainers are of the second kind, and more extreme: stops of $3$ and $-1/2$, a mean of $5/4$ and a temperature of $7/4$. So the record is now held by a shape that is not a fair fight at all, and the temperature is high because the gap between what the two players can achieve is large rather than because both stand to gain.

That is a mechanism a reader can look for, and it makes a mild prediction: the next record will be more lopsided again, because a region gets hot most cheaply by being good for one player and terrible for the other. Nothing here tests that, and it is the sort of thing a size-twelve sweep would settle in passing.

What the plateau was #

Read as a sequence of rises rather than as a curve, the hottest temperature goes up at sizes four, six, eight and eleven — gaps of two, two and three.

The rung below saw the first three gaps and read a quarter every two sizes, which was the only reading available and was wrong at the next term. One gap of three is not a pattern either, and this page claims no rule: what it claims is that the sequence rises irregularly and that a sweep ending at a fixed size will read whatever plateau it happens to stop inside as a ceiling.

Five of the rung below’s claims can be scored and one survives. The containment is real; the ceiling, the pattern of rises and the coldness of the extra material are all wrong.

It is worth being fair about the rung below’s own hedging. It proposed the containment as a test rather than as a finding, and named the eleven- and twelve-square regions as the way to run it. The test has been run, it passes, and it does not do the work it was proposed to do. That is a better outcome than the test failing, because a containment that holds and explains nothing is a fact about these shapes that the next rung has to account for.

The four shapes, looked at #

The four attainers are worth a paragraph on their own, because they are drawn in the figures and a reader may want to know what to look for.

Each is an eleven-square region with an eight-square hot core and three more squares, and the three are not attached in an obvious way — they are not a tail hanging off the core, and the cores overlap inside the region rather than sitting in one corner of it. The two orbits are mirror images: one worth $\{3{\ast} \mid -1/2\}$ and one worth $\{1/2 \mid -3{\ast}\}$, which is the first negated, and each orbit holds two shapes because each is asymmetric.

The values are worth reading. A temperature of $7/4$ comes from stops of $3$ and $-1/2$: Left moving first can reach $3$ and Right can hold the position to $-1/2$, so the fight is worth three and a half and the temperature is half of that. That is a very lopsided position — the mean is $5/4$ — and it is a different shape of hot from the eight-square attainers, three of which are worth $\{3/2 \mid -3/2\}$ and are symmetric.

So the step from $3/2$ to $7/4$ is not the same object getting bigger. It is a differently-shaped fight, and that is one more reason to distrust the extrapolation the rung below made from three terms.

Why a sweep reads a plateau as a ceiling #

The failure mode here is general and it is worth naming, because this site keeps meeting it.

A sweep to size $n$ produces a sequence of $n$ terms and a reader reads its tail. If the tail is flat, the two available readings are the sequence has stopped and the sequence is on a plateau, and no amount of looking at the same $n$ terms distinguishes them. The only cure is another term, and another term is the expensive thing — eleven squares is 34,053 shapes and forty-six seconds, and twelve is roughly four times as many with a costlier evaluation on each.

So the rung below could not have known, and the phrasing it used — the ceiling — was a stronger word than the evidence carried. What would have been available is the count of attainers: eight shapes at size eight, fourteen at nine, forty-three at ten. A quantity whose attaining set is growing is not a quantity that has stopped; a bound reached by more and more shapes is a bound being approached from below, not a wall. That reading was available on the rung below’s own data and nobody made it.

The values that keep arriving is the other page on this site about a sequence that looked finished, and the two together are the site’s account of what a flat tail is worth.

What the pictures can carry here #

Unusually for this ladder, the figures on this page draw the object. A Domineering region is a set of squares, so a table of regions drawn as grids is the thing itself rather than a summary of it, and a reader can see the four eleven-square attainers and the eight-square shapes inside them.

What a reader cannot see is the containment. Five hot cores inside one eleven-square region, overlapping each other, is not a thing a static grid shows: it would need the core highlighted inside the region, five times, or an animation. The tables give the count of hot cores instead — five, on each of the four — which is the honest fallback and is a much weaker picture.

The other thing not drawn is a thermograph. Every temperature on this page comes from one, and a reader wanting to see why a particular eleven-square region is worth $\{3{\ast} \mid -1/2\}$ has to take the value on trust or go to reading a thermograph. That is the ordinary division of labour on this site — the region pages draw regions and the temperature pages draw walls — and it is worth naming when a page needs both.

What this does not settle #

Twelve squares is not here. Eleven took forty-six seconds and twelve is out of reach by enumeration on this machine. So whether $7/4$ is itself a plateau or a step is exactly the question this page has just asked of $3/2$ and cannot answer — and a reader should treat seven quarters at eleven as a measurement rather than as a new ceiling.

The four attainers are not explained. They are named, drawn and shown to contain the hot eight-square cores, and nothing here says what makes them hot. The rung below’s five eight-square shapes have the same standing: witnesses without an argument.

And the containment is not a theorem. Four regions each containing five hot cores is a fact about four regions. Whether every hot region contains a hot smaller one is a much stronger statement, and the way to test it is to run the containment check on the forty-three ten-square attainers rather than on the four at eleven — which is affordable and is not done here.

One size is one size. Everything on this page rests on a single new term of a sequence, and a single term is what the rung below had too much confidence in. The claim made here is deliberately narrower: not the sequence rises for ever, but the sequence rises at eleven, so the eight was not a ceiling. A negative claim needs one counterexample and this is one; a positive claim about what happens next would need what nobody has.

Regions, not boards. Every shape here is a connected region, which is what a Domineering board falls into once play has separated it. One fight makes a board a fight is why a board’s temperature is its hottest region’s, and it is what licenses asking about regions in isolation at all. A board of eleven free squares in one piece is a rare thing in play; what a game actually produces measures how rare.

The attaining-count signal is a hypothesis about four numbers. Eight, fourteen, forty-three, four is a suggestive sequence and it is four terms of a sequence with nine, and the counts below the plateau are 1, 1, 3, 4, 10 — which do not obviously fit many means about to be exceeded. So the signal proposed at the end of this page is proposed rather than measured, and the measurement is cheap.

Normal play, and temperatures from the ordinary thermograph.

What is left of the rung below #

It is worth ending on what the correction does not undo, because the rung below did substantial work and most of it stands.

Its census is right. Every temperature it reported for sizes three to ten is the temperature this sweep finds, and the eight witnesses at size eight are the eight this page draws. Its finding that three of the five attaining orbits are the hot core of a position on every board it swept is untouched, and it is the reason the eight-square shapes matter at all — they are shapes a played game produces, not curiosities of an enumeration.

What is wrong is one word. Ceiling claims a bound and the data supported a plateau, and the difference is only visible from one size further on. That is the ordinary way a sweep-based claim fails, and the ordinary fix is the one applied here.

One number for a player #

Nothing on this page is a strategy, and it is worth saying what the closest thing to one is.

A player looking at a Domineering board wants to know where the fight is, and playing the hottest says to move in the hottest region. This ladder has been asking how hot a region can be, which bounds how wrong a player can go by misjudging one.

The bound has moved: a region of eleven free squares can be worth $7/4$ to move in, where before eleven the largest known was $3/2$. That is a quarter of a move, and a quarter of a move is enough to change which of two components to play in. So the practical content of this correction is small and real: a player estimating a large region’s stake from the eight-square shapes inside it will under-read it, and the under-reading grows with the region.

Where the ladder goes next #

The cold anchor has eight 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, what the ceiling is a fact about, and now that it was not a ceiling.

The rung above is the attaining count. This page’s best diagnostic was one nobody used: the number of shapes attaining the maximum, which grew from eight to fourteen to forty-three across the plateau and fell to four at the step. If that is a general signal — a maximum attained by many shapes is about to be exceeded, and one attained by few has just been reached — then it reads a plateau off a single size, which is what the whole difficulty here is. Testing it costs nothing new: the attaining counts are already in the sweep at every size, and the sizes below the plateau are enough to say whether the pattern holds before the plateau as well as across it.

Two neighbours are worth the trip. Eight squares, and no hotter is where the five witnesses and the plateau were found, and it is the page this one corrects — its measurements all stand and its word for them does not. And how hot a real position is is where the temperatures of positions a game actually reaches were measured, and it is worth reading beside a page about the hottest regions there are, which a game may never produce.