Puzzle ZOIV 

Workings

Hint
How many 1 x 1 squares are there, how many 2 x 2, …?

Answer
2,870 squares.

Reasoning
We have squares sizes from 1 x 1 up to 20 x 20, and these are the counts:
 1 x  1 = 400
 2 x  2 = 361
 3 x  3 = 324
 4 x  4 = 289
 5 x  5 = 256
 6 x  6 = 225
 7 x  7 = 196
 8 x  8 = 169
 9 x  9 = 144
10 x 10 = 121
11 x 11 = 100
12 x 12 =  81
13 x 13 =  64
14 x 14 =  49
15 x 15 =  36
16 x 16 =  25
17 x 17 =  16
18 x 18 =   9
19 x 19 =   4
20 x 20 =   1

Total = 400 + 361 + 324 + 289 + 256 + 225 + 196 + 169 + 144 + 121 + 100 + 81 + 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 = 2870.