Every essay — page 31
Temperature
How much is at stake, measured. Thermographs, cooling, and why a player moves where the game is hottest.
One fight makes a board a fight
The rung below found 16 per cent of the components a played game produces to be hot, against 53 per cent of the catalogue they are drawn from, and predicted that the share of hot boards would be much larger. Taking the same play-outs and tallying at the board gives 32 per cent — twice the piece figure and not ten times it, because a Domineering board carries only 1.68 pieces and the hot ones cluster on the same boards.
The quantity that does not order a board
The rung below found the players leaving an environment at the larger of a position's two temperatures, and proposed that a board should therefore be played in the order of that quantity. Over 220 boards of three components it plays exactly on 124 against playing-in-the-hottest's 196, loses 85 of the 97 disagreements, breaks Hotstrat's guarantee on six boards, and costs nine points on its worst one.
A rule that beats the hottest
When promoting a component by its answer's temperature failed, the reverse was proposed — discount it instead — with a prediction, made before the sweep, that it would not beat playing in the hottest component. It does. It plays exactly on 201 of 220 three-component boards against 196, wins two thirds of the boards where the two disagree, keeps inside a guarantee proved for the other rule, and the gap widens as the board grows.
The obstacle was the catalogue
The rung below could not measure the early game because its regions were too large for the catalogue, and asked for a bracket rather than a value. No bracket is needed: a twelve-square region evaluates in five milliseconds and an eighteen-square one in under a second. What was expensive was cataloguing every shape rather than sweeping the positions a board actually reaches — and the sweep says a board is hot four times in five three moves in, and cools when it breaks up.
The two numbers at the top
The rung below found the crossover of a sente fight to be its temperature less half its answer's, and said a proof would settle the depth question with it. The depth question is settled without the proof, by construction: group the positions by their two top temperatures and the crossover is single-valued on every group, however far apart the third temperature is — and the formula survives a fourth level of fight, which the rung below never reached.
The worst value in its own interval
The rung below scored a component by its temperature less its hottest answer's and asked what rate the answer should really be charged at. Every weight strictly between nought and one scores the same and beats the rung below's choice of one at every board size — because a ranking rule's score is a step function of its own coefficient, and one is exactly where two components tie.
Eight squares, and no hotter
The rung below found no Domineering position hotter than three halves on four boards and asked for the position that attains it. It is a region of eight squares, there are five of them up to symmetry, three are the hot core of an attaining board on every size swept — and the ceiling holds at nine and ten squares too, where the obvious extrapolation predicted seven quarters.
The premises an induction would need
The rung below settled by a grouping test that a position's crossover depends on its own temperature and its answer's and on nothing below them, and asked for the induction. The four paragraphs are not written here; the checking they would rest on is. The law holds at five levels, survives translation, heating and cooling — and none of that is the step.
A pool built to have an answer
The coefficient in the rule score a component by t − λa scored identically for every λ in the unit interval, because the rule reads an ordering and that pool's orderings changed at three places. A pool designed to have twelve crossings turns the interval into thirteen different rules, and all three board sizes agree on one cell: between a quarter and a third.
Which top is the top
The crossover law's proof rests on the walls above the crossover being governed by the top two options, and the check was never run. Run on 23,586 heights it holds exactly — but only when the options are ranked by mean value. Ranked by the temperatures the law is stated in, it fails on a fifth of them.
The ceiling was a plateau
Three halves of a move looked like a ceiling on a Domineering region's temperature: it held at eight squares, at nine and at ten, and the rise that had been a quarter every two sizes stopped. At eleven squares four regions reach seven quarters — and they contain the hottest eight-square shapes and are hotter than them, so the extra material is not cold.
A description, and not a detector
The rung below noticed that the count of shapes attaining the hottest temperature grew across a plateau and collapsed at the step, and proposed it as a way to read a plateau off a single size. The growth is exact — five plateaus, no exception — and the rule is impossible: five orbits precede a rise at seven squares and no rise at eight.
The four paragraphs prove something else
Three rungs earned the right to write the crossover law's proof as two straight walls meeting where the law says. The walls are straight — one of them everywhere, for a trivial reason, and the other only above the answer's own temperature. The crossover sits below that height, so the geometry holds nowhere the law is about, and where it does hold it proves the temperature instead.
A second pool, designed differently
One designed pool put the rule's best coefficient between a quarter and a third, and all three board sizes agreed. A second pool, built by the identical greedy criterion from different material, has no cell that is best at every size — so the coefficient is a property of the pool and there is no number to find.
The residues as a sequence
Four of the eight residue sequences never repeat, and a recurrence is not a description. There is a closed form and it is not for the game: the stops and the temperature of the n-th residue are periodic with period one, two or four on every sequence in the pool, while three of them produce a different game at every n.
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