NL-NWO 1968 Problem 5

A square of side $n$ ( $n$ natural) is divided into $n^2$ squares of side $1$ . 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 $8$ rectangles of width $n - 1$ . (b) Determine the number of rectangles with width $n -k$ ( $0\le k \le n -1,k$ integer). (c) Determine a formula for $1^3 + 2^3 +...+ n^3$ .