Combinatorial Game Theory
who wins, and by how much
Series

Misere cost — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Classes needed, as the heaps get bigger — Dawson's chess ·137. How many kinds of position there are, against how large a heap the universe allows. Under normal play the answer stops growing as soon as the Grundy values stop growing. Under misère play it does not stop, and every new class is a pair of positions that behave identically under normal play and differently under misère.

    The cost is in the closure, not in the positions

    Under normal play, Dawson's chess needs four classes for every heap up to twelve, because its Grundy values stay at three or below there. Under misère play the same game needs six, then twelve, and the number rises with the universe rather than with the position — which is a different kind of expense entirely.

    part 1 · complexity
  2. One misère outcome, searched. The number of positions a misère search visits to decide the outcome of a sum of k heaps of Dawson's chess, each heap at most 9, on a logarithmic scale: the average over the sums and the worst single sum, for k from one to eight. Normal play decides the same sums from 20 stored values.

    A misère sum is searched, not added

    Under normal play the outcome of a sum of heaps is a nim-sum of numbers already known: twenty stored values decide every sum of Dawson's chess with heaps up to nine, however many heaps it has. Under misère play each sum is a new position to search. One outcome costs six positions for a single heap, two hundred for four heaps and over five thousand for eight, and a table of every eight-heap outcome costs a hundred thousand. The misère quotient is the only thing that brings the price back down.

    part 2 · complexity
  3. The closure that is enough. A grid for Dawson's chess with heaps up to 9: rows are the largest positions classified, from one heap to four; columns the largest tests, from none to five heaps. Each cell is the number of classes found. The counts stop growing at two-heap tests and three-heap positions.

    Two heaps of testing are enough

    A misère quotient is computed by testing positions against positions, and the universe used to find twelve classes of Dawson's chess was every position of up to four heaps tested against every other — 511,225 outcomes. Varied one size at a time, the count stops growing at tests of two heaps and positions of three: 12,100 outcomes find the same twelve classes. The narrower universe the earlier essay drew did not merge anything; it held fewer positions. And the corner that is enough moves: for Kayles at heap twelve, two-heap tests miss a class.

    part 3 · complexity
  4. 12 classes, 7 questions. A grid for Dawson's chess with heaps up to nine: rows are the 12 misère classes of positions of at most four heaps, columns the 7 tests a greedy search chose, and each cell the outcome — N for the player to move, P for the other — when the test is added to the class.

    Twelve classes, seven questions

    Twelve misère classes of Dawson's chess were found by testing 715 positions against 715 others. Seven of those tests are enough to tell every class from every other — a greedy choice against a floor of four, since each test is one yes-or-no question. Kayles needs nine of 715 and Nim sixteen. The seven cost almost nothing to use and cannot be found without the whole closure, and they do not carry: the tests found with heaps up to seven tell apart only seven of the twelve classes with heaps up to nine.

    part 4 · complexity
  5. A staircase, not a slope. The number of misère classes of Dawson's chess positions as the largest heap allowed rises from three to 16, computed with positions of at most three heaps and tests of at most two. The count stays flat for several heaps at a time and then jumps.

    A staircase, not a slope

    With the misère closure cut forty-fold, Dawson's chess can be classified at heaps far beyond nine. The count of classes is a staircase: six from heap three to eight, twelve from nine to twelve, seventeen from thirteen to sixteen. Normal play steps once in that range, from four to eight at heap thirteen, where a Grundy value of four first appears. Misère play steps there too, and once more at heap nine, where normal play does not move at all — the first wild heap. Heaps eleven, fifteen and sixteen are also wild and move nothing.

    part 5 · complexity

All series