Combinatorial Game Theory
who wins, and by how much
The collection

Every essay — page 32

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

Values

What a position is worth — numbers, and the things that are not numbers, and how to find the simplest one.

The order one day out. Whether the values born by day three still form a lattice. Twice as many pairs are incomparable as at day two, and every incomparable pair still has a least upper bound and a greatest lower bound — so the order becomes more tangled without becoming ragged.

One of four questions

Three rungs of this ladder rest on sweeps of day two — 22 values, 253 pairs. Day three is 1,474 values and over a million pairs, and only one of the four questions can be asked of it. The order can: twice as many pairs are incomparable and every one of 1,606 sampled still has a least upper bound and a greatest lower bound, none of them a value day two already had. The other three compare sums of day-three values, which are born on day six, and sixty of those exhausted an eight-gigabyte heap.

6 figures · Lattice

All ladders · Every object named here · The position index · Figures that play back · Search