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no of rectangles in n x n square

Source: Netherlands - Dutch NMO 1968 p5

January 31, 2023
geometryrectanglecombinatoricscombinatorial geometry

Problem Statement

A square of side nn (nn natural) is divided into n2n^2 squares of side 11. Each pair of "horizontal" boundary lines and each pair of "vertical" boundary lines enclose a rectangle (a square is also considered a rectangle). A rectangle has a length and a width; the width is less than or equal to the length. (a) Prove that there are 88 rectangles of width n−1n - 1. (b) Determine the number of rectangles with width n−kn -k (0≤k≤n−1,k0\le k \le n -1,k integer). (c) Determine a formula for 13+23+...+n31^3 + 2^3 +...+ n^3.
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