Combinatorial Game Theory
who wins, and by how much
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The thread: Small things decide — page 2

Infinitesimals are smaller than every positive number and are not zero. In a close game they are the whole margin, which is why the theory bothers with them.
Not a domination, in the order the rung below meant. The second pass's deletions scored as dominations in two orders: the partial order on games, and the order on stops. Sums and comparison

Not a domination, in that order

The rung below asked which pair the second reduction acts on, taking for granted that the operation is a domination. It is not: on none of the 525 deletions is a surviving option greater than or equal to the deleted one. In the order the reduced form actually works in — both stops at least as good — every deletion with a survivor is a domination, the dominator is unique on all but twelve, and it always comes from the other part.

 Hold the tail, vary everything in front. The grouping test: every tail against every prefix, asking whether the rate at which a widening gap stops mattering is a function of the tail alone. Particular games

Two strips that end the same way

If a Push strip's sensitivity is governed by its last run, then two strips agreeing at the far end should behave the same however different their fronts. On 78 of 80 tails they do, exactly. On two of them a single empty square in the prefix reaches across a gap that grows without bound and halves the rate — and the run reading turns out to be sound in one direction only.

 One of the two quantities is inert. The candidate predictors of a pre-period's length, each scored by correlation against the measured length. Impartial games

The quantity that carried nothing

The rung below proposed predicting a pre-period's length from the saltus and the period. The saltus correlates with it at −0.03, which is nothing; the period correlates at 0.77 with a coefficient of one, so a pre-period is about one period long. The digit that predicts whether there is one predicts nothing at all about how long.

 Chomp to 12 × 8: one needle on every bar but one. Every Chomp rectangle up to 12 columns by 8 rows, with the number of winning opening moves in each cell, found by search. All but one have exactly one; the 10 × 8 bar has 2. Cells in blue belong to the families whose winning move can be stated in a sentence — a single row, two rows, or a square; cells in gold are found only by searching. Out in the world

Where the needle has a sentence

Strategy stealing proves the first player wins every Chomp bar and names no square to take. On two families the square can be said in a sentence — a square bar and a bar two rows deep — and in both the sentence is a pairing that names every later move too. Three rows deep the needle wanders, and the observation that every bar has exactly one needle survives ninety-four rectangles and fails on the ninety-fifth.

 What the stops settle. The four open entries of the outcome table over every sum of two day-two values, read by the four inequalities on the stops of a sum. The stops settle 176 of 369 and are never wrong; the other 193 all have a bound exactly on nought. Sums and comparison

What a part would have to report

Outcome classes leave four entries of the addition table open. Give each part its two stops as well and four inequalities settle 176 of the 369 open day-two sums, never wrongly — and every sum they leave has a bound sitting exactly on nought. Used as a lookup table instead, even the whole thermograph with the outcome beside it leaves 96 sums undecided. Four outcomes do what no thermograph can: who wins a day-two position alone, beside −1, beside 1 and beside a star names it, and no three questions do.

 The residue of a sum is the hotter part's. Every hot day-two value added to every hot day-three value, 7,854 sums in all, sorted by what happens to the residues: when the parts differ in temperature, on all 4,712 such pairs, the sum's residue is the hotter part's alone; when they are equally hot the residues add on 880, the sum grows colder on 2,237 and cancels to a number on 25. 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.

 Colder exactly when the residues cancel. Equally hot pairs of hot values, 3142 from days two and three and 3000 sampled from day three, sorted by whether the sum stays as hot, grows colder or is a number, and by whether the two residues cancel. Every sum that stays as hot has residues that do not cancel, and every sum that grows colder or is a number has residues that do. 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.

 A number from the mast does not finish the job. Eight reports a part could make, from its two stops to its value, with the number of sums no lookup table indexed by the report can decide. Beside day three: stops 19,036, stops and outcome 16,402, stops and who is to move when each ends 17,088, the atomic weight of each foot 14,924, the feet 5,842, the feet with who chooses 4,962, and with who is to move 3,164; the value leaves none. On day two the atomic weight leaves 159, more than the 136 left by the stops and the outcome. Sums and comparison

A foot is a menu, not a number

When a sum's fight ends level on nought, the winner is decided by what is left at the foot of each part's fight, and the obvious number to report from there is the atomic weight. Over 32,428 sums of a day-two and a day-three value it leaves 14,924 undecided, and on day two it does worse than a single outcome letter. What sits at a foot is a choice, made by one player, with someone to move when it is made. Report all three and every sum of two cold parts is decided; 3,164 sums are still not, and each holds a hot part whose own fight is not the one played in company.

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