Combinatorial Game Theory
who wins, and by how much
Series

Entailing — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Top Entails, one heap at a time. Each heap with the outcome of playing it alone, the Grundy value an ordinary solver would give it, and the moves that win from it. Taking the top coin of a heap forces the opponent to answer in that heap, which is a kind of move an ordinary sum of games has no room for.

    A move that must be answered

    Every argument about sums assumes the parts are independent: a move in one leaves the others alone, and the reply may go anywhere. Top Entails denies it — take the top coin of a heap and the opponent must answer in that heap. The nim-sum then misreads 9 of 36 two-heap positions, and two heaps of two coins are a first-player win, which no impartial game the theory covers can be.

    part 1 · limits
  2. 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.

    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.

    part 2 · limits
  3. The arrow reversed. Values of heaps 1 to 10 in entailing Nim and Top Entails, read off their pairings with Nim heaps, when an entailing move compels the reply in the same heap and when it compels it in a different heap. Pointed elsewhere, entailing Nim's heaps are worth their size, as in Nim, and no heap of either game is loony.

    A compulsion that points elsewhere

    Top Entails and entailing Nim are decided by one value more than Nim because the compelled reply stays in the heap that compelled it. Point it at a different heap instead and the two games part company. Entailing Nim does not notice — its heaps are worth exactly what Nim's are, because Nim's winning reply was never in the heap just moved. Top Entails loses its loony heaps and then every value of the Nim kind: no assignment of nimbers explains its sums, because a heap of three beside two heaps of one is a win and beside two heaps of two is a loss.

    part 3 · limits

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