"Any positive integer can be written as the sum of four square numbers", says Lagrange's four-square theorem. After years of intermittently trying to figure out * why*, a recent peek reveals why the

- a clever-deep algebraic formula to reduce the challenge to prime numbers
- a pigeonhole principle to reduce it to multiples of primes
- an infinite descent to reduce that multiple to 1

Q.E.D. — and how non-obvious, in retrospect!