How many squares (of any size) are there on a chess board ?
Note that there are squares of size 1, 4, 9 etc. So we need to calculate total number of squares (of all possible sizes).
Also expand your solution to the total number of rectangles on the chess board.
A chess board has small 64 squares, but there are other squares which can be formed using small squares,
Squares can be formed of 2*2 size (using 4 small squares)… other squares which can be formed using 9 small squares (3*3), 16 small squares (4*4) and so on..
The key point is to find the number of squares for each combinations. To find the squares, we can think of positions where a square starts (say, top-left position). If we consider squares of smallest size, there there will be 8 horizontal positions (top-left corner of each square in the horizontal row) and 8 vertical positions (top-left corner of each square in a vertical column).. hence the total number of positions of smallest (single) square is
8*8 = 64
Similarly, we can find the larger squares (of sizes 4, 9, 16, 25, 36, 49 & 64)
Hence the total number of squares will be:
Hence, the total number of squares on a chess board is = 204.
For, total number of rectangles, let me write a new post ( may be tomorrow or day after).