Combinatorial Game Theory
who wins, and by how much
Generator

Nim-multiplication below 16, and every field axiom checked

Nim-multiplication below 16, and every field axiom checked
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 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.

5 essays call nimber-table. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

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. Impartial games

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.

 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. Impartial games

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.

 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. Particular games

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.

 What a finite closed company is made of. The finite closed companies found by the search, counted by the properties they share. Every one of them consists of games equal to their own negatives and has a size that is a power of two, and not all of them are made of nimbers. Where it stops

The company that is closed

Restricted equality licenses substitution only inside a company closed under addition, and none of the five companies this site computes in is closed — day two keeps a quarter of its own sums. Searching for companies that are closed finds seven, at one, two, four and eight members, and every member of every one of them is its own negative.

 The second closure picks out the nimbers. The seven finite companies closed under addition, tested for closure under forming options. The four that are groups of nimbers keep every option; the three containing plus-or-minus one lose theirs. Where it stops

The closure that picks the nimbers

Closure under addition lets a sum be rewritten and turned out to admit companies that are not nimbers at all. Closure under forming options lets a subposition be rewritten, and it pulls the other way: every company this site computes in has it and none has the first, and among the seven finite addition-closed companies, keeping every option is exactly being a group of nimbers — four of seven, both directions, no exception. Demand both at once and nineteen of twenty-two day-two values generate nothing finite.

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