Combinatorial Game Theory
who wins, and by how much
Generator

The same position, and two rulers to measure it with

The same position, and two rulers to measure it with
The same position, and two rulers to measure it with. Nim positions with the length of their input under two encodings — the heap sizes in binary, and the counters themselves — beside the work the nim-sum does. The work never changes. Which of the two lengths it is compared against decides whether the same algorithm reads as linear or as exponentially fast, and hardness claims are always made against one particular encoding.

Nim positions with the length of their input under two encodings — the heap sizes in binary, and the counters themselves — beside the work the nim-sum does. The work never changes. Which of the two lengths it is compared against decides whether the same algorithm reads as linear or as exponentially fast, and hardness claims are always made against one particular encoding.

1 essay calls encoding-cost. 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

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

PositionWorth OutcomeDrawn in
heaps 1, 1 0 P Nim is easy, in binary · Misère play · Tame and wild · Two misère outcomes are not enough · What a tame heap may be replaced by · What a value leaves out · What a wider pool rescues
heaps 1, 1, 1 ∗1 N A pass is not a move · Nim is easy, in binary · Misère play · Nim, and the nim-sum · Tame and wild · The patch that generalised · The sentence that solved the other convention · The theorem that needed none of the theory · Two misère outcomes are not enough · What a tame heap may be replaced by
heaps 1, 2 ∗3 N Nim is easy, in binary · Misère play · Two misère outcomes are not enough
heaps 1, 2, 3 0 P A pass is not a move · A token on a graph · The move that gives counters back · Nim is easy, in binary · Misère play · Nim, and the nim-sum · Tame and wild · The patch that generalised · The sentence that solved the other convention · The theorem that needed none of the theory · Three players and no answer · What a value leaves out · What a wider pool rescues
heaps 10, 20, 30 0 P Nim is easy, in binary
heaps 100, 200, 300 ∗384 N Nim is easy, in binary
heaps 1000, 1000, 1000 ∗1000 N Nim is easy, in binary
heaps 1000, 2000, 3000 ∗3968 N Nim is easy, in binary
heaps 1000000, 2000000, 3000000 ∗3932160 N Nim is easy, in binary
heaps 1024, 1024, 1024 ∗1024 N Nim is easy, in binary
heaps 1048576, 1048576, 1048576 ∗1048576 N Nim is easy, in binary
heaps 16, 16, 16 ∗16 N Nim is easy, in binary
heaps 2, 3 ∗1 N Nim is easy, in binary · What a tame heap may be replaced by
heaps 256, 256, 256 ∗256 N Nim is easy, in binary
heaps 3, 4, 5 ∗2 N Nim is easy, in binary
heaps 3, 5 ∗6 N Nim is easy, in binary
heaps 4, 4, 4 ∗4 N Nim is easy, in binary
heaps 5 ∗5 N Nim is easy, in binary
heaps 5, 5 0 P Nim is easy, in binary · What a value leaves out
heaps 5, 5, 5 ∗5 N Nim is easy, in binary
heaps 5, 5, 5, 5 0 P Nim is easy, in binary
heaps 5, 5, 5, 5, 5 ∗5 N Nim is easy, in binary
heaps 64, 64, 64 ∗64 N Nim is easy, in binary
heaps 7, 11, 13 ∗1 N Nim is easy, in binary

Where it is called

Changing this generator changes every one of these figures.

The same position, and two rulers to measure it with. Nim positions with the length of their input under two encodings — the heap sizes in binary, and the counters themselves — beside the work the nim-sum does. The work never changes. Which of the two lengths it is compared against decides whether the same algorithm reads as linear or as exponentially fast, and hardness claims are always made against one particular encoding. What it costs

Nim is easy, in binary

Three heaps of a thousand counters take thirty bits to write down and three thousand counters to lay out. The nim-sum does three exclusive-ors either way. Whether that counts as fast depends entirely on which of the two numbers the work is compared against.

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