Symmetry Breaking & WLOG

Order the variables before you start.

If a statement is symmetric, you may assume $a \ge b \ge c$ without loss of generality. That one line often halves the casework and unlocks rearrangement and Chebyshev, both of which need a known ordering.

The discipline: state the assumption explicitly and check the problem really is symmetric (or that you have handled the other orderings). A false WLOG is one of the most common ways a correct-looking solution loses marks.

For cyclic-but-not-symmetric expressions you may only rotate, not reorder — a distinction graders watch for.

Where it appears on Lemma: Level 5 onward, in nearly every inequality and combinatorics proof.

Train it on Lemma

Symmetry Breaking & WLOG unlocks at Level 5 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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