Combinatorial Game Theory
who wins, and by how much
Series

End-Nim — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. End-Nim: a player at each end of the row. Rows of heaps in which Left may take from the leftmost heap and Right from the rightmost. The value beside each row was computed by the game recursion and reduced to canonical form; the outcome beside it says who wins. A single heap is a Nim heap, because both players may take from it — and that is the last thing about this game that looks like Nim.

    Taking from the ends

    End-Nim is Nim's board with a player at each end, and it takes one sentence to state. Not one of its 5,460 small positions is worth a non-zero number — the game is all-small, so zero is the only number any of them can reach — and there are 2,693 distinct values between them. The outcome says a great deal more: 4,738 of those positions are won by the same player whoever moves, and on two heaps the rule is that the larger end wins.

    part 1 · positions
  2. How much of End-Nim is a Nim heap. Rows of End-Nim by length, with the share worth a nimber beside the share that are palindromes and the number of distinct values. The impartial share falls from all of the one-heap rows to a fifteenth of the six-heap rows, while the values multiply.

    Where the nimbers run out

    A single End-Nim heap is a Nim heap and every palindromic row is worth a nimber, so the impartial theory looks as though it might get a long way into a partizan game. It gets one row in thirteen. Five nimbers occur in five and a half thousand rows, the palindromes account for two fifths of them, and the rows worth something else run to 2,693 distinct values.

    part 2 · positions
  3. Three counts, and two of them are the same number. Rows of End-Nim that read the same backwards, rows equal to their own negative, and rows worth a nimber. The second and third counts are identical over every row in the sweep, so being its own negative is exactly the condition for being worth a nimber in this game.

    The rows that are their own mirror

    Four hundred and ten End-Nim rows are worth nimbers and 168 of them are palindromes, so a condition covering the other 242 was outstanding. It is that the row is equal to its own negative — and on this game that condition is not merely sufficient but exact, which is more than the group law promises and is a fact about End-Nim rather than about games.

    part 3 · positions
  4. Two thousand values and no temperature. Every End-Nim row of up to six heaps, with its temperature. Not one of the 5,460 is hot, every one of them is worth an infinitesimal, and the 361 worth numbers are 361 rows worth nought.

    A game with nothing at stake

    The rung below explained a coincidence with a claim it did not compute: that End-Nim carries no hot self-negative values. The census says something stronger. Not one of 10,919 rows across three shapes of board is hot, every one of them is worth an infinitesimal, and the 361 rows the earlier census called numbers are 361 rows worth nought. The reason is one line of the rule, and 2,693 distinct values sit underneath it.

    part 4 · positions

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