1009.21 – Squares on a Checkerboard

How many squares are on or inside a checkerboard? (Hint: More than 64.)

Solution

Here is a tally of the squares.

1×1:642×2:493×3:364×4:255×5:166×6:97×7:48×8:1\begin{aligned} 1 \times 1 &:& 64 \\ 2 \times 2 &:& 49 \\ 3 \times 3 &:& 36 \\ 4 \times 4 &:& 25 \\ 5 \times 5 &:& 16 \\ 6 \times 6 &:& 9 \\ 7 \times 7 &:& 4 \\ 8 \times 8 &:& 1 \end{aligned}

The total number is 204. Interesting numbers, eh?