Each new layer is the next odd number

A patch is 4 tiles across. You want it 5 across. You do not start again.

Rule: Growing a square adds an L-shaped layer. The layers are the odd numbers in order. Add them from 1 and every total is a square.

• 1 + layer 3 = 4
• 4 + layer 5 = 9
• 9 + layer 7 = 16
• 16 + layer 9 = 25

Each layer is two equal strips and one corner tile. Two equal strips make an even number. The corner tile makes it odd.

This gives a shortcut. Take 75² = 5625. To reach 76, add two strips of 75 and a corner. That is 2 × 75 + 1 = 151. So 76² = 5625 + 151 = 5776.

This gives a test too. Take away 1, 3, 5 and 7 from 16, step by step.

• 16 − 1 = 15
• 15 − 3 = 12
• 12 − 5 = 7
• 7 − 7 = 0

You reach 0, so 16 is a square. Try 20. After 1, 3, 5 and 7, 4 is left. The next odd number, 9, is more than 4. So you never reach 0, and 20 is not a square.

Tap Add a layer. Watch each new L-shape.