Combinatorial Game Theory
who wins, and by how much
Ladder

Thermograph — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 012345601234valuetemperaturetemperature 2mean 3Left's wallRight's wall{5 | 1} — mean 3, temperature 2

    Reading a thermograph

    A thermograph is two walls rising from a number line, closing in as the tax on moving increases, and meeting where the position stops being worth fighting over. Everything about a position's hotness is in the shape.

    rung 1 · temperature
  2. 3 independent regions, in order of what is at stakeregionsettles atat stakein the account{4 | 0}22+ 2{3 | 1}21− 1{2 | 0}11+ 1means 5, alternating 2 — the account predicts 7the recursion says Left moving first gets 7the account is exact here — every region is a simple fight between two numbers

    The endgame, accounted for

    Add up what each region is worth, then add the biggest thing at stake, subtract the next, and so on down. On a board of simple fights the result is exact — and the moment one region has a fight inside it, the account is out by a point.

    rung 2 · temperature

All ladders