Counting Squares in a SquareHow many grid squares are there in a grid square of size N×N? Obviously, there is the N×N square itself and N² small, 1×1 squares. But, for N, say equal to 5, there are also 2×2, 3×3, and 4×4 squares. In general, for the N×N square there are smaller squares of all possible sizes, starting with 1×1 all through N×N. How do you count those? The applet bellow suggests a way of counting M×M squares contained in an N×N square,
 |Activities| |Contact| |Front page| |Contents| |Algebra| |Store| Copyright © 1996-2012 Alexander BogomolnyThe position of an M×M square is fully determined by its upper left 1×1 piece. (The upper left corner is chosen for convenience. Any other small square inside a bigger one determines uniquely the position of the latter.) Inside the N×N square this 1×1 piece may occupy a limited number of positions. These form a square of side (N - M + 1) in the upper right corner of the N×N square. It follows that the number of M×M squares inside the N×N square equals (N - M + 1)×(N - M + 1).
The total number T of grid squares within the N×N grid is the sum:
the sum of the first N squares. This we know to be
|Activities| |Contact| |Front page| |Contents| |Algebra| |Store| Copyright © 1996-2012 Alexander Bogomolny |
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