Bouton's invariant, checked over 512 positions

Nim positions in binary, one column per bit. Bouton's 1901 argument is that a position is a loss for the mover exactly when every column holds an even number of marks — and that from such a position every move breaks a column, while from any other position some move repairs them all. Both halves are checked here over every position in the range rather than illustrated once, and the middle row shows the repairing move being made.

1 essay calls boutons-proof. 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.