11th Iberoamerican 1996 Problem A1 Find the smallest positive integer n so that a cube with side n can be divided into 1996 cubes each with side a positive integer. Solution
: 13. Divide all the cubes into unit cubes. Then the 1996 cubes must each contain at least one unit cube, so the large cube contains at least 1996 unit cubes. But 12 3 = 1728 < 1996 < 2197 = 13 3 , so it is certainly not possible for n < 13. It can be achieved with 13 by 1.5 3 + 11.2 3 + 1984.1 3 = 13 3 (actually packing the cubes together to form a 13 x 13 x 13 cube is trivial since there are so many unit cubes). 11th Ibero 1996 © John Scholes jscholes@kalva.demon.co.uk 22 Oct 2000