Combinatorial Game Theory
who wins, and by how much
Generator

Welter positions and what they are worth

Welter positions and what they are worth
Welter positions and what they are worth. Coins on a strip, with the Grundy value the recursion returns and the nim-sum the squares would have if they were independent heaps. The two columns are the essay: they hardly ever agree.

Coins on a strip, with the Grundy value the recursion returns and the nim-sum the squares would have if they were independent heaps. The two columns are the essay: they hardly ever agree.

2 essays call welter-coins. 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.

The positions it draws

19 distinct positions, harvested by running this generator again at the options each essay passed it.

PositionWorth OutcomeDrawn in
0, 1 0 P No two heaps alike
0, 1, 2 0 P No two heaps alike
0, 1, 4, 5 0 P No two heaps alike
0, 2, 4, 6 0 P No two heaps alike
0, 2, 5, 7 0 P No two heaps alike
0, 3 ∗2 N No two heaps alike
0, 3, 4 0 P No two heaps alike
0, 6 ∗5 N No two heaps alike
1, 3, 5 0 P No two heaps alike
1, 3, 6 ∗7 N No two heaps alike
1, 4 ∗4 N No two heaps alike
1, 4, 6 0 P No two heaps alike
2, 3 0 P No two heaps alike
2, 3, 8, 9 0 P No two heaps alike
2, 4, 5 0 P No two heaps alike
2, 5 ∗6 N No two heaps alike
3, 7 ∗3 N No two heaps alike
4, 5 0 P No two heaps alike
6, 7 0 P No two heaps alike

Where it is called

Changing this generator changes every one of these figures.

Welter positions and what they are worth. Coins on a strip, with the Grundy value the recursion returns and the nim-sum the squares would have if they were independent heaps. The two columns are the essay: they hardly ever agree. Impartial games

No two heaps alike

Welter's game is Nim with one extra clause — no two heaps may be the same size — and the clause is fatal to the nim-sum, which gives the right answer in none of the 120 three-coin positions. What replaces it is a function of pairs: ⟨a | b⟩ = (a ⊕ b) − 1, exact on all 55 two-coin positions, and nim-added over every pair it is exact on the whole board provided the number of coins is even.

 Four readings, one game. The closed form, the strict mating, the cancelling matching and the parity term, with how much of the game each accounts for. Impartial games

The pairing the formula hides

Welter's closed form sums a function over every pair of coins and needs an extra term when the count is odd, which the rung below called a surprise. Read as a matching it is not: an odd number of coins cannot be paired, the left-over coin contributes its own square, and some matching gives the value on every position measured.

The whole library · The position index · The figures that play back