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Counting Squares

Problem

A standard chessboard measures 8 x 8 squares, making sixty-four 1 x 1 squares. But there are many more squares, for example 3 x 3, 4 x 4, etc.


How many squares are there altogether on a chessboard?


Solution

There are, 64 times (1x1), 49 times (2x2), 36 times (3x3), 25 times (4x4),
16 times (5x5), 9 times (6x6), 4 times (7x7) and 1 times (8x8) squares.

Making 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204 squares in total.

How many squares are there on a n x n grid?
What about rectangles on an m x n grid?

Problem ID: 107 (Mar 2003)     Difficulty: 1 Star

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