Combinatorial Game Theory
who wins, and by how much
Series

Partizan subtraction — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. What each heap is worth. The value of a single heap of each size. Nothing here repeats: the forms grow deeper as the heap grows, which is what stops the impartial theory's periodic table from having an analogue.

    Two players, two lists

    Give each player their own list of how many counters they may take and the impartial theory stops applying. What survives is the outcome: it settles into a repeat, for every pair of lists, and that is a theorem. What does not survive is the value — on four of six pairs swept it has no repeat inside sixty heaps, and the birthdays are still climbing at the edge of the window.

    part 1 · positions
  2. A sequence with a rule and no period. The values of the subtraction game with Left taking 1 or 2 and Right taking 1 or 3, from heap 5 up. Each is the game whose only Left option is nought and whose only Right option is the value three heaps below — checked at every heap rather than asserted, and the two heaps where it fails are the two below the seeds.

    A sequence with a rule and no period

    The values of the subtraction game where Left takes one or two and Right takes one or three never repeat — thirty-one heaps, thirty-one different values. They are nonetheless completely described: three seeds and the rule v(k + 3) = {0 | v(k)} generate every one of them, which is what a pattern without a period looks like.

    part 2 · positions
  3. Two conditions, one of which survives. Two candidate conditions on a pair of subtraction lists, scored over all 961 pairs drawn from one to five. Translation holds on 83 pairs and every one of them repeats; all-odd holds on 49 and four of them do not.

    The condition that survived the wider sweep

    Which pairs of subtraction lists have a value sequence that repeats? Over the 49 pairs drawn from one, two and three, two conditions answer it identically — a translation and all-odd — and both are exactly right. Over the 961 pairs drawn from one to five, all 83 translations still repeat with no exception and four all-odd pairs do not, at heap ninety with a period as long as forty-two. Neither condition is necessary: 104 pairs repeat that satisfy neither.

    part 3 · positions

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