Combinatorial Game Theory
who wins, and by how much
Series

Additivity — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Knowing who wins is not enough. Three pairs of positions, every one of which is in outcome class N on its own. Their sums are not all the same, and not all in the same outcome class — so the outcome of a sum cannot be worked out from the outcomes of its parts, and that is why the theory needs values.

    Outcomes do not add

    Knowing who wins each part of a position tells almost nothing about who wins the whole. Counted over every sum of two values born by day two, six of the outcome table's ten entries are settled and four are not — and every settled one is settled by the order rather than by anything about outcomes. Two first-player wins reach all four classes between them.

    part 1 · sums
  2. 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.

    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.

    part 2 · sums
  3. 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.

    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.

    part 3 · sums

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