Combinatorial Game Theory
who wins, and by how much
Ladder

Disjunctive sum — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 2+-1+1/4+∗={5/4 | 5/4}outcome Leach sprig is a separate game; a move is a move in one of themthe total was computed by adding the games, not the labels

    The sum is the object

    Real positions come apart into independent regions, and a move happens in exactly one of them. That operation — the disjunctive sum — is what the whole theory is built to survive, and it is the reason values exist at all.

    rung 1 · sums
  2. ↑ + ∗worth ↑∗, outcome NLeft to movemove inleavingthe whole position becomesverdict↑0∗loses∗0↑winsexactly one of the 2 moves wins, and it is in ∗each verdict is the outcome of the whole position after the move, computed rather than judged

    Which part to move in

    The value of a sum is the sum of the values. The move in a sum is not the move in any part, and there is no rule that reads it off the values — in the smallest interesting example, the only winning move is in the component worth nothing.

    rung 2 · sums

All ladders