Complete Coverage Strategy

Intro to Proofs · Axiom Academy

When a claim has only finitely many cases, you can prove it by the most direct move there is: check every single one. A claim is only proven when every case is covered Here is a claim about lining up three tiles, each red or blue: at least two of the three tiles must share the same color. Try one arrangement and it holds. Try another, it holds again. But a handful of working examples is not a proof — maybe the very next arrangement is the one that breaks it. With only three tiles there are exactly 2 × 2 × 2 = 8 arrangements in all, so we can settle it for good: check all eight. Watch the proof get built. A marker sweeps across all eight arrangements; each one it reaches is verified and the coverage bar climbs. Notice what happens when a case is skipped — leave even one uncovered and you have proven nothing. Only when the bar reads 8 / 8 is the claim actually proven. Proving by checking every case is called proof by exhaustion (or exhaustive case analysis): enumerate all the possibilities, verify each, and the claim is settled. Test the claim on a single arrangement Click any tile to flip its color (or pick a color, then click a tile). The verdict updates instantly: does this arrangement have two tiles of the same color? Try to build one that breaks the claim — three tiles, two colors, see if you can dodge a repeat. Any line-up of 3 tiles (each red or blue) must contain at least two tiles of the same color.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.