Problem of the Week 1071

Pack Them In

As a Valentine's Day present, Alice got Bob a set of infinitely many cubes, having side-lengths 1, 1/2, 1/3, ..., one cube of each side. But how to gift-wrap them? Alice knows that the harmonic series diverges, so she cannot stack one on top of another in finite space. But she also knows that the total volume of the cubes is finite, since

1 + 1/2³ + 1/3³ + ... < 1 + 1/x³ dx = 1.5.

So the total volume is less than 1.5 and it seems as if she could just pack them all into a gift-box of dimensions 1 × 1 × 3/2. At this point she realized that a better idea would be to give Bob the cubes and the gift box and make him do the packing.

Show how to arrange the cubes so that they fit into a 1 × 1 × 3/2 box.

© Copyright 2007 Stan Wagon. Reproduced with permission.