How many rectangles can be made out of a 3x3 box of squares?

Number of rectangles that can be made out of an [math]n[/math] x [math]n[/math] grid is given by [math]\Sigma k^3[/math] from 1 to n. read more

The Total No. of rectangles (not squares) can be formed from 3x3 - box 1 unit height x 2 units length (in one row=2nos. & in 3-columns)= 2x3 = 6nos. 1 unit height x 3 units length = 1x3 = 3nos. read more

Thus, the number of rectangles in a 5x5 square is the sum of the 1 square wide rectangles in the 1x1, 2x2, 3x3, 4x4, and 5x5 squares or 4 + 18 + 48 + 100 = 170. Therefore, for the typical chess board problem of 8x8 squares, we have a total of 4 + 18 + 48 + 100 + 180 + 294 + 448 = 1092. read more