Combinatorial Game Theory
who wins, and by how much
Ladder

Numbers — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 011/2{0 | 1}031{0 | 3}-570{-5 | 7}142{1 | 4}1/411/2{1/4 | 1}the marked point is the value; the hollow one, where it differs, is the midpoint

    The simplicity rule

    When both players' options are numbers, the position is worth the simplest number strictly between them. Not the midpoint, not the average, and the difference between "simplest" and "middle" is the entire content of the rule.

    rung 1 · values
  2. day 00day 1-11day 2-2−1/21/22day 3-3−3/2−3/4−1/41/43/43/23day 4−5/2−7/4−5/4−7/8−5/8−3/8−1/81/83/85/87/85/47/45/2each new number is the simplest one in a gap — which is the simplicity rule, applied everywhere at once

    The day a number is born

    Start with a position in which neither player can move, apply one rule, and the numbers appear — but only the fractions with a power of two underneath, and only in a particular order. That order is what "simplest" means.

    rung 2 · values
  3. {2 | 0} + 1worth 3 | 1, outcome L — Left wins whoever movesLeft moves in the fight{2 | }worth 3outcome Lstill a win for LeftLeft moves in the number{2 | 0}worth 2 | 0outcome Nnow whoever moves next wins — and it is Right's turnthe number was worth a whole point and moving in it cost the gamea number is never the right move while anything that is not a number is available

    Numbers avoid numbers

    In a position with a number in it and anything else, the number is never the right move. That is a theorem rather than a heuristic, and it is the closest this subject comes to advice a player can carry into a real game.

    rung 3 · values

All ladders