There are 55 squares in a 5 by 5 grid.
There are 25 squares of one grid.
There are 16 squares of four grids i.e. 2 by 2
There are 9 squares of nine grids i.e. 3 by 3
There are 4 squares of sixteen grids i.e. 4 by 4
There is 1 square of twenty-five girds i.e. 5 by 5
Hence, there are total 25 + 16 + 9 + 4 + 1 = 55 squares.
You must have noticed one thing that total number squares possible of each size is always a perfact square i.e. 25, 16, 9, 4, 1
For a grid of N by N, the possible number of squares are
= N2 + (N - 1)2 + (N - 2)2 + (N - 3)2 + ......... + 32 + 22 + 12
For 1 by 1 grid, total squares = 12 = 1
For 2 by 2 grid, total squares = 22 + 12 = 5
For 3 by 3 grid, total squares = 32 + 22 + 12 = 14
For 4 by 4 grid, total squares = 42 + 32 + 22 + 12 = 30
For 5 by 5 grid, total squares = 52 + 42 + 32 + 22 + 12 = 55
