The Domino Effect

Intro to Proofs · Axiom Academy

A row of falling dominoes is the whole idea behind mathematical induction — push the first one, make sure each knocks the next, and infinitely many fall. How do you prove a statement is true for every whole number — 1, 2, 3, and on forever — when you can't possibly check them one at a time? Line up dominoes instead. If the first one falls, and every domino is close enough to knock over the next, you don't have to watch each one: the whole row is guaranteed to go down. Watch the perfect chain. The dominoes are evenly spaced, so each falling tile reaches the next. Press play to push the first one and let the cascade run to the end. Base case (the first falls) plus each-knocks-the-next (the spacing) means the cascade never stops — that is exactly mathematical induction. What if one domino can't reach the next? Now widen the gap between one domino and the next, then push the first tile. If a falling domino can't reach its neighbor, the cascade stops dead right there. In a proof this is the step "if statement number k is true, then statement number k+1 is true" — break that link and everything past it stays standing. Close the gap until every tile reaches the next and the chain completes — that unbroken link, repeated for every k , is the inductive step.

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