Q: How many squares are on an 8 x 8 checkerboard? 10 x 10? How many rectangles?

8 x 8 There are:

64 1 x 1

49 2 x 2

36 3 x 3

25 4 x 4

16 5 x 5

9 6 x 6

4 7 x 7

1 8 x 8 for a total of 204 squares

For 10 x 10

100 1 x 1

81 2 x 2

64 3 x 3

49 4 x 4

36 5 x 5

25 6 x 6

16 7 x 7

9 8 x 8

4 9 x 9

1 10 x 10

for a total of 385 squares

The rectangle counting is significantly more challenging.

Click here for a beautiful explanation from Nigel Coldwell