Cooling by 1, and heating back
Each row is a position, its temperature, what it becomes when every move is taxed, and what comes back when the tax is refunded. The refund is not an inverse: a position whose temperature was below the tax has already frozen into a number, and heating a number does nothing at all.
20 essays call
chill-table. The drawing above is what it returns with no arguments at all; every
call below passes it something, because a placement that passes nothing draws whichever
member of the family the generator happens to default to rather than the one its essay argues
about.
The positions it draws
40 distinct positions, harvested by drawing the figure again at the options each essay asked for.
Where it is called
Every one of these figures is a drawing of the same construction, so a correction to it corrects them all.
Temperature
Cooling by exactly one
Cooling is usually met as a way of reading a thermograph: a tax, and two numbers at the height of the tax. Applied as an operator it returns a position instead — and then the obvious question is whether heating gives the position back. Over the 1,474 values born by day three, 27 survive the round trip and 15 of those are numbers that never moved. Cooling throws away almost everything it touches.
Temperature
A number and a fight
Charge a position exactly what it is worth fighting over and the fight disappears, leaving the mean value — with something still attached to it. Over all 1,122 hot values born by day three the residue is smaller than every positive number, it is a star in 942 of them, and it is never nothing. So a hot game is its mean plus a fight plus a remainder that no number reports, and the remainder is what decides close games.
Temperature
Cooling adds and heating does not
The two operators are presented as a pair, and they are not one. Cooling a sum is the same as cooling the parts and adding, on every one of the 1,768 pairs tried, at two taxes and on two pools. Heating fails on 263 — and not for the obvious reason: in every failure neither part and not the sum is a number, so the clause exempting numbers never fires at the top. It fires two levels down, where an option of a sum is one part's option plus the whole of the other.
Temperature
The operator chosen for one game
Chilling is cooling by exactly one, and the one is not derived from anything. It is chosen because Domineering mostly runs at that temperature — and measured against this site's whole Domineering catalogue it turns thirteen of fifteen boards into numbers or numbers plus a star, and warms thirteen of the fifteen back exactly. They are not the same thirteen: eleven boards do both, two freeze too far to be recovered, and two stay hot and come back on the nose.
Sums and comparison
What is left when the small change is thrown away
Canonical form answers a demanding question: which positions are interchangeable inside every sum whatever. A player with a hot board does not have every sum — an infinitesimal difference cannot decide anything against a genuine fight — so there is a coarser question with an exact answer. The reduced canonical form takes the 1,474 values born by day three to 61, with 292 of them collapsing to zero, and it is a homomorphism on all 8,100 pairs tested only when a second pass is made.
Temperature
Below zero
Temperature is described as urgency and urgency has no obvious bottom, but the scale stops at −1 and only the numbers are on it. The rung above is exactly zero, and the 337 values born by day three that sit there are the ones a number cannot be told from — flat thermograph, nothing at stake, and an infinitesimal that no number can see. Two independent computations agree on the classification for all 1,474.
Temperature
The operator that puts the star back
Chilling is not invertible: it freezes, and 400 values born by day three collapse onto 29. Both heating and Norton's warming operator are exact right inverses of it — each lands back where it started, on all 400 — and they pick different preimages, differing on 396 of them and differing by exactly a star on 335. The clause that separates them is one line long and it is about the integers.
Temperature
How hot a day gets
A day of construction buys exactly one degree of temperature — nought, then one, then two — and the value attaining the maximum is unique on every day: ∗, then {1 | −1}, then {2 | −2}. The distribution underneath is not tidy at all: it peaks at a half, leans to the right of the peak, and has a hole in it at one and three quarters where nothing is born.
Temperature
How hot a real position is
Counted one value at a time, a tenth of the subject is hot. Counted one position at a time — every board this site has enumerated, all 11,397 of them — it is a twentieth, two thirds of the positions are worth numbers outright, and ten of the seventeen rulesets never produce a hot position at all.
Temperature
What a game actually produces
Fifty-three per cent of the Domineering regions of at most eight squares are hot. Of the components eleven hundred random games actually produce, sixteen per cent are — and ten per cent once single squares are counted. The figure is the same on three sizes of board, so it is a property of play rather than of the board, and it says that every temperature census this site has taken over a catalogue overstates how hot the game is by a factor of three.
Temperature
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.
Temperature
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.
Temperature
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.
Temperature
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.
Temperature
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.
Temperature
The hotter residue survives
Charge a hot position its own temperature and subtract its mean, and an infinitesimal is left — its residue. Add two hot positions and the means add, the temperatures do not, and the residues do something stranger than either: on every one of 4,712 pairs whose parts differ in temperature, the residue of the sum is the hotter part's residue and nothing of the colder's. Cooling is additive, and a part cooled past its own temperature is a bare number. Residues add only when two parts are exactly as hot, and even then the fights usually cancel into something colder. The summary forgets; the game does not.
Temperature
Colder exactly when the residues cancel
Add two fights of one temperature and the sum either stays that hot or grows colder, and which one happens is decided by the smallest thing either fight carries. The sum grows colder exactly when the two residues are negatives of each other — on all 3,142 equally hot pairs from days two and three and 3,000 sampled from day three, and by a two-line argument from cooling's additivity. What survives the cancellation is another matter. Group the colder sums by their parts' thermographs and the survivor differs in every group that holds more than one pair: {1 | 0} and {1 | 0, ∗} draw the same diagram and leave different infinitesimals behind. The diagrams fix only the survivor's temperature: nought, or exactly the lowest bend of either part.
Temperature
Why cooling has to add
Cooling came through a sweep of 1,768 sums untouched while heating failed 263 times, and the sweep's own account was cautious: cooling is not additive in general, and a richer pool would break it. Aimed exactly where that account said to look — infinitesimals hidden under low temperatures, at taxes of a quarter and a half — 107,816 more pairs break nothing, because there is nothing to break. Cooling distributes over every sum of short games. Its freezing clause fires on one side of the equation and not the other in 6,032 of those pairs, and costs nothing every time, because what it hands back is the mean, and means add.
Temperature
The bonus an integer is owed
Heating fails to distribute over a sum where cooling never does, and the repair proposed for it was to change what its exception returns: heat a number into a switch instead of leaving it alone. Run on the same 1,768 pairs, that repair fails on every one of them, beginning with nought plus nought. What adds is a heating that changes which numbers it exempts and pays the bonus on the moves an integer holds in reserve — Norton's product, and heating with no exception at all, which at a whole-number bonus is the same operator. Given heating's exception, cooling stops adding too.
Temperature
An exemption that costs nothing below one
Heating that leaves integers alone fails to add at a bonus of one and adds on everything at a bonus of a half, and the natural guess was that a half is where it starts to go wrong, since that is where a fraction's moves become worth fighting over. It is not. Across twelve bonuses nothing fails at any value below one, and dozens of pairs fail at one exactly. The reason is the simplicity rule: below one, the recursion sends every integer to itself, so the exemption is not doing anything at all.
The whole library · The position index · The figures that play back