Hot game — where it appears
Named by 47 essays across 5 fields — each of them below, with the objects they name alongside it.
What is at stake
Some positions both players are desperate to move in, and some neither player wants to touch. The difference is a number — how much the move is worth — and it turns out to be the most useful single quantity for deciding where to play.
One board, two rules
Col forbids painting next to your own colour. Snort forbids painting next to your opponent's. One word differs, the boards are identical, and the values that come out are not the same kind of object.
Numbers avoid numbers
In a position with a number in it and anything else, the number is never the right move. That is a theorem rather than a heuristic, and it is the closest this subject comes to advice a player can carry into a real game.
Two hot fights that add to a cold number
The mean of a sum is the sum of the means, every time. The temperature is not — it is bounded by the hottest part and is often far below it. Two positions each worth fighting over can add to a plain number that neither player wants to touch.
The same fight, eight times over
The mean is not roughly what a position is worth. It is the number that eight copies of the position stay close to — and the theorem is that the closeness does not decay as the copies pile up. The gap stops at the temperature, and stays there for ever.
Worth nothing, and worth fighting for
A switch is a position both players want to move in. Its average value can be zero while the difference between getting there first and second is enormous, and that gap is a second number every position carries.
Big is not the same as hot
A player sizes a move by how much changes hands when it is played, which is the number in every endgame book. The theory sizes it by temperature. On a plain switch the two agree exactly, so nothing shows; on a move with a follow-up they come apart, and the count gives up more than the guarantee the theory's rule carries.
Counting at the end changes everything
Go is scored. So are Dots and Boxes, chess and almost everything anybody plays for money — and none of them is the kind of game this site's whole apparatus is built for. The simplest scoring game there is shows what that costs — the normal-play theory gives every position of it the same answer, and the answer is useless.
Sente is a fact about the rest of the board
A move that must be answered is called sente, and the word is used as though it described the local position. It does not. The same fight is answered while the rest of the board is quiet and ignored once it is busy, and the crossover — measured by solving the whole board at every temperature — sits at the follow-up's own temperature.
A thermograph is built from its options'
The diagram is not measured, it is computed — each wall is an extremum over the options' opposite walls, shifted by the tax. That construction is why a hot follow-up lowers the temperature instead of raising it, and why two fights with the same swing can differ by a factor of two in what is at stake.
An environment made of coupons
Beside the game sits a stack of coupons worth 4, 3, 2, 1, 0, and a player may always take the top one instead of moving. Play the whole thing out and two quantities the theory computes are measured instead: {4 | 0} comes out worth exactly 2, its mean value, and the coupons stop at 2, its temperature. For {10 | {9 | 1}}, whose temperature is 1, they stop at 7/2 — because what the stopping coupon measures is the hottest temperature anywhere in the tree.
One rule makes it cold, the other hot
Col forbids painting next to your own colour and Snort forbids painting next to your opponent's. Over all 540 colourings of six graphs, not one Col position has anything at stake and 61 Snort positions do — the difference between a game that is a count and a game that is a series of fights, produced by one word.
A thermograph with two bends
A wall bends where the option holding it up stops holding it up, and most drawn thermographs bend once. { {5 | {3 | 1}} | 0} bends twice, at 1 and 3/2, and neither height is its temperature of 7/4 — every bend lies strictly below where the walls meet. Over a census of 17,255 hot positions two levels deep, a second bend in one wall never happened once.
Where the fight stops
Keep taking the biggest thing on offer until somebody is left facing a number, and the number reached is a stop. Every position has two of them, one for each player moving first, and between them they say how a fight ends — except for the part of it no number can reach.
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.
What a move is worth to the player making it
The gain from a move is the option minus the position it was played from — and that is a game rather than a number, so two moves can be incomparable instead of one of them being best. Temperature is what happens when the largest of those games is asked for a single number.
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.
The switch a player is imagining
Every account of a hot position ends up as "worth about m, and worth t to move in", which is the switch {m+t | m−t}. For a plain fight that summary is the position exactly. For a fight with anything behind it the leftover is not a rounding error — on one position here it is a whole second fight of temperature two.
The same strip without the jump
Delete one clause from Toads and Frogs — the hop over an opponent — and the game is Elephants and Rhinos. Over the same 3,279 strips the values do not become simpler in the way a reader would guess: every value that is a number becomes an integer, against 172 fractions with the jump in, and the count of positions worth fighting over nearly doubles. Removing a move made the game hotter.
Two games in one environment
A coupon stack measures a position: play the whole board out and the coupon the players stop at is the temperature, the score is the mean. Put a second position beside the first and one of the two measurements stops working. Over 36 ordered pairs the contributions still add to the means every time, and the coupon a fight is entered at moves on 13 of them — without either position changing.
A rule with a guarantee
Evaluating a sum of a dozen fights is impossible; following a rule is not. Move where the stake is largest, and over 220 sums of three hot components the rule scores exactly what perfect play scores in 196 of them, is never more than one point behind, and never ends more than the largest single stake below the mean. The rule that is supposed to be different — answer the threat — chose differently in none of the 220.
What a number does to a fight
Adding a number to a position moves everything and changes nothing: over three thousand translations the temperature never once shifted and both stops moved by exactly the number added, every time. What the number decides is whether the fight is worth having — and the interval where the two players are confused is exactly the open interval between the negated stops, right in all 2,890 cases away from its endpoints and wrong in 110 that are all on them.
When a switch is not a switch
A position {a | b} with numbers on both sides is a fight only while a is above b. Sweeping the boundary with a fixed at 2 and b climbing from −2 to 3 turns up three regimes rather than the two the definition suggests: eight fights whose temperature is exactly half the gap, one position at a = b that is 2∗ and is not a number, and beyond that numbers chosen by the simplicity rule — which on 8 of 12 sampled cases is not the midpoint.
The fight never runs backwards
Left's stop is never below Right's — in every one of 1,780 distinct values, computed twice by two independently written routes, with nothing that disagreed anywhere. The inequality is what makes a mean value well defined and a fight a fight; and where it collapses to equality, 433 of the 460 cold positions turn out not to be numbers at all.
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.
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.
A rule with no promise at all
Playing in a hottest component comes with a bound: never more than the largest single temperature below the mean of the board. Over 220 sums the bound holds 220 times — and so does the bound for a rule with nothing behind it, which scores exactly what perfect play scores on 205 sums against the hottest rule's 196. The control that shows the bound is doing work is the rule that plays the coldest component, which breaks it 74 times and loses up to eleven points.
Nobody wants to move here
A position is a number exactly when every move loses ground for the player making it. The test never mentions numbers, it disagrees with the ordinary one on none of the 1,474 values born by day three — and the reason a position fails it is not that somebody wants to move. It is that somebody cannot afford to wait.
Topple it from either end
A row of blue and red is the picture this site opens with, and under Hackenbush's rules it is always a number. Knock the pieces over instead of cutting them — everything on the chosen side falls — and 480 of the 510 rows up to eight pieces stop being numbers. The two games agree on sixteen rows, every one of a single colour, and the temperature of the hottest row climbs by exactly a half for each domino added.
How cold a sum of hot games can be
The temperature of a sum is at most the largest temperature in it, and the bound leaves the whole interval below it open. Over 1,035 pairs the sums do not use that interval: 864 sit exactly at the maximum, 160 are frozen outright, and eleven land anywhere in between — every one of them with a component whose wall bends.
What a move nobody makes is worth
If Right's move in a fight has to be answered, then Left's move in the same fight prevents an exchange Right was going to get for nothing. That is a reverse sente, and pricing it is the awkward case: over sixty measurements the gain matches the local temperature once and the follow-up's swing twice, and the band over which the move is worth taking is not the band over which the move it reverses is sente.
What is left when the copies pair off
A pile of n copies stays within a bounded distance of n times the mean, and the distance never grows. The difference is a game rather than a number, and what it actually is has a much better answer: for a plain switch it alternates between one fight and nothing at all, and for a fight with a follow-up it is different every time — bounded in size and unbounded in complexity.
When two thermographs can be added
The temperature of a sum is not the sum of the temperatures, and the natural repair is to add the whole diagrams instead. Over every pair of hot values born by day two the added walls always bound the true ones and the mast always comes out right — and the whole diagram is right exactly when at most one of the two components is hot, which is precisely the case a reader has no use for.
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.
How hot a background has to be
The reduced canonical form throws away infinitesimals, and the rung below asked for a bound: how hot must the rest of the board be for the discarded part not to matter? There is no such bound. Safety is not monotone in the background's temperature — an eighth is safe, a quarter is not, two is safe again — and the quantity that does decide it is not a temperature but a stop.
A fight with no midpoint
The mean of {a | b} is the midpoint and the temperature is half the gap — on the twenty-one values of day three where a and b are numbers. One hundred and forty-six others have the same shape and not the hypothesis, and the repair that suggests itself, reading the two stops instead of the two options, holds on about three quarters of them and no more.
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.
The answer that starts another fight
A local move is answered while the ambient temperature stays below the follow-up's — and that rule, which this site has carried since the anchor opened, is exact only when the answer ends the fight. When the answer starts another one the crossover is exactly half the follow-up's temperature, on every position tested, and a third level of fight does not halve it again.
Which shapes are worth fighting over
Forty-four of the 104 Domineering regions of at most six squares are worth numbers and the rest are not, and the rung below said no visible property of a shape predicts which. Half of that is wrong: a region only one orientation fits in is a whole number, on all eleven of them, for a reason a reader can supply in a sentence. The other half stands, and thirty-three shapes are what makes it stand.
Counting the moves each side has
How many dominoes could each player still place? Subtract, and there is a whole number computable from the drawing with no game theory in it. Over 1,042 regions it is the value on 141 of the 315 worth numbers, lands between the stops on 619 of the other 727, and its failures are two different kinds — one of which was inevitable and one of which is a fact about the game.
The bend above the top
The chain reading gets three values in 2,403 wrong because it counts bends that the diagram never reaches. Counting only the bends below the position's own temperature fixes all three and breaks none — the first exact reading on this ladder, and it needs one comparison rather than the envelope the rung below expected.
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.
An environment instead of a stack
The guarantee behind playing the hottest component survives one step down a board's sorted temperatures and fails at two. Against a coupon environment as hot as the board it survives all of them — because the rule stops being approximate and starts being optimal, on every sum in the pool built to punish it.
The second bend is the boundary
Adding two thermographs wall by wall gives a diagram that is right at the mast and wrong below it. Over every pair of hot values born by day two, the added walls sit outside the true ones at every height — an outer envelope with the truth somewhere inside — and the two pictures separate at exactly the lower of the two temperatures, on all twenty-eight pairs. Above that height both components are still fights and the addition is exact; one sixteenth below it, every pair has parted.
What has to break before a pawn is worth a number
Every value the blocked-file model can hold is an infinitesimal, and the reason is one sentence about the move rule rather than anything about pawns. Break that sentence — a pawn stuck behind a friend, a square only one side can hold — and integers, switches and positions worth fighting over arrive at once.
A hypothesis has to hold all the way down
Milnor's bound is proved by induction over the play, so the condition it needs has to hold at every position the play can reach. Checked on the row instead, ninety-two pairs pass the test and twenty-four of them break the bound. Checked at every subposition, twenty-eight pairs pass and none breaks it.
Named alongside it
The objects these essays reach for when they reach for this one.
TemperatureMean valueThermographSwitchExhaustive searchDisjunctive sumStopsInfinitesimalCold gameCanonical formCoolingStar (∗)