Nim — the series · Combinatorial Game Theory

Nim with heaps of 3, 5, 7. Heaps of counters; a move takes any number from one heap. The position is a loss for the player to move exactly when the binary digits of the heap sizes cancel in every column — the nim-sum — and that is the whole of the theory of Nim.

Nim, and the nim-sum

Three heaps of counters, take as many as you like from one of them, and the player who takes the last counter wins. The winning condition is not a search, not a table, and not a heuristic — it is the bitwise exclusive-or of the heap sizes, and it was found in 1901.

part 1 · impartial

Poker Nim from 3, 5, 7, with reserves of 4 and 4. Nim with one extra kind of move: a player may put any number of counters back onto a heap from a private reserve. It looks as though a losing player could stall for ever. They cannot, and the winner is decided by exactly the same nim-sum as ordinary Nim — checked here over every position within a stated range rather than argued.

The move that gives counters back

Poker Nim adds one rule to Nim — a player may put counters back onto a heap from a private reserve. It looks as though a losing player could stall for ever. The winner is decided by exactly the same nim-sum, and the reason is the single most useful idea in the whole reduction apparatus.

part 2 · impartial

Nim-multiplication below 16, and every field axiom checked. The nim-product, defined by taking the least value the product is not forced to be — the same manoeuvre as the mex rule, applied to a product rather than to a move. The result is that these values are not merely a group under nim-addition but a field: every axiom is checked over the whole table here, including an inverse for every non-zero value, and the sizes at which the axioms fail are reported rather than avoided.

The nimbers multiply

Nim-addition is exclusive-or and everybody meets it first. There is also a multiplication, defined by the same take-the-least-value-not-forced manoeuvre as the mex — and it makes the nimbers below sixteen a field, with every axiom checked here and an inverse for every non-zero value.

part 3 · impartial

A grid of coins, and a multiplication table. The Grundy values of a two-dimensional coin-turning game, computed from its own move rules by a mex at every cell. Down the left and across the top are the one-coin values of the two one-dimensional games it is built from. Every cell is the nimber product of its two edge values — the multiplication defined for the nimber field on algebraic grounds — and beside the grid are the three combining rules a reader would try first, each killed on a named cell.

The tartan theorem

The nimbers are a field, with a multiplication defined by a mex-style rule that looks like an algebraist's amusement. Lay two coin-turning games on a grid and the Grundy value of each square is the nimber product of its two coordinates — which is the point at which the multiplication stops being a curiosity and starts computing answers.

part 4 · impartial

What the theorem replaces. Every grid a brute-force solve can reach, valued both ways, with the grid the theorem is normally drawn at underneath. Twelve squares is four thousand arrangements against twelve products; sixty-four squares is eighteen quintillion against sixty-four.

Four hundred and seventy steps

The tartan theorem replaces a search with a multiplication. Measured on every grid a brute-force solve can reach, the two agree on all of them — and the ratio doubles with every square added. On the 8 × 8 grid the theorem is normally drawn at, the search would have to value eighteen quintillion arrangements; the theorem needs twenty-six different nimber products, and computing all of them by the rule that defines them looks at four hundred and seventy pairs.

part 5 · impartial