Combinatorial Game Theory
who wins, and by how much
Theme

The thread: One clause decides it — page 5

Change a word of the rule and the values change completely. A cliff or a wall, a jump allowed or forbidden, a pass that may or may not end the game — the same board, and nothing in common.
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. Impartial games

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.

 What each assumption buys back. Forty-five positions of up to two heaps of up to eight, sorted into classes that no test position of up to three heaps can tell apart, under four rules. Two-player Nim has sixteen classes, both three-player conventions have forty-five, and one player against a coalition of two has six. Where it stops

What survives a coalition

Three-player Nim has no winner without a convention and no sum theory at all. Fix one player against the other two and both come back: every position has a winner, the winner is read off the number of heaps of one modulo three and the number of larger heaps, and forty-five positions fall into six interchangeable classes — fewer than Nim's sixteen. Fix a scoring rule instead and the winner comes back, identical to one of the conventions, and nothing else does: no two positions are interchangeable.

 A nimber or loony. The value of every heap from one to ten in Top Entails and in entailing Nim, read by pairing each heap with every Nim heap. Even heaps are loony in both games; odd heaps are nimbers, nought or star in Top Entails and ∗(h + 1)/2 in entailing Nim. Where it stops

One value more than Nim

Top Entails broke the nim-sum: nine of thirty-six two-heap positions came out wrong when a move could compel the reply. The repair is one extra value. Read each heap off the heap itself, and every heap of Top Entails and of a second entailing game is either a nimber or loony — a win for the mover in any company. One rule then decides every sum of those games and Nim together: 4,016 positions, and it is never wrong. A loony heap is a pass, and the compulsion is what buys it.

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