Lemma › Techniques › Coloring Arguments
Olympiad problem-solving technique
Coloring Arguments
Paint the board so the impossible shows its color.
Color the cells with two or more colors so that every legal piece or move interacts with the colors in a fixed way — then count. The mutilated chessboard falls in one sentence: each domino covers one black and one white, but 32 black and 30 white remain.
Beyond two colors: stripes for pieces, four-colorings for T-tetrominoes. The color count is just an invariant wearing paint.
Train it on Lemma
Coloring Arguments unlocks at Level 4 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 78 lessons, 624 curated problems and unlimited generated practice at six difficulties. Free to start — no card, no trial clock.
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