Squares and odd numbers

Look at the gaps between one square and the next: 4 − 1 = 3, 9 − 4 = 5, 16 − 9 = 7, 25 − 16 = 9. The gaps are the odd numbers, in order.

Turn that round: adding the odd numbers one by one, starting from 1, gives the squares in turn.

It must start from 1: 13 + 15 + … won't work.

The book shows why with a picture instead of words. Start with one tile. To grow the square by one on each side, wrap an upside-down L around two of its edges. Each L is exactly 2 tiles longer than the one before, which is why you add 1, 3, 5, 7, … 'In mathematics, sometimes arguments and reasoning can be presented without any words. Visual proofs can be complete by themselves.'

So: the sum of the first n odd numbers is n². Every square is a sum of odd numbers in a row, starting from 1.

That gives a test. Keep subtracting odd numbers from 25: 25 − 1 = 24, 24 − 3 = 21, 21 − 5 = 16, 16 − 7 = 9, 9 − 9 = 0. It lands exactly on 0, at the 5th step, so 25 = 5².

Try 38: 38 − 1 = 37, − 3 = 34, − 5 = 29, − 7 = 22, − 9 = 13, − 11 = 2, − 13 = −11. It jumps past 0, so 38 can't be made this way and is not a perfect square.

Two more useful facts:

• the nth odd number is 2n − 1: the 1st is 1, the 2nd is 3, the 6th is 11, the 36th is 2 × 36 − 1 = 71;
• the whole numbers strictly between two squares p and q number q − p − 1. Between 16² = 256 and 17² = 289 there are 289 − 256 − 1 = 32 (that's 2 × 16: between n² and (n+1)² there are always 2n numbers).