Combinatorial Game Theory
who wins, and by how much
Series

Cram — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Cram on 4 by 4: the pairing strategy. Cram is Domineering with the orientations shared: either player may place a domino either way up, so both players have exactly the same moves and the game is impartial. Every position therefore has a Grundy value, and this board's was computed by the mex rule over its own placements.

    Cram

    Domineering with one word of the rule changed: both players may place a domino either way up. That makes the game impartial, and the entire partizan apparatus collapses into a single Grundy value — on the 4 × 4 board, Domineering's canonical form runs to 114 characters of nested braces and Cram's answer is the one character 0.

    part 1 · impartial
  2. Misère Cram, board by board. The normal-play value and misère genus of every Cram rectangle with at least two rows and at most twenty squares. 12 are fickle, 4 × 4 is firm, and 3 × 6 and 4 × 5 are wild.

    Cram with the last move losing

    Under misère play every Cram rectangle to twenty squares is tame at the start except 3 × 6 and 4 × 5, and twelve of the fifteen boards two rows deep behave like one counter or none. The pairing that settles even boards under normal play hands the misère game to the first player, who wins by any vertical domino. And the tame boards are not tame inside: wild positions appear on boards as small as 3 × 4, always at ten empty squares, the length at which the strip goes wild.

    part 2 · impartial

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