South African Mathematics Olympiad

We must decide whether the books are to be packed flat, on their spines, or upright (or possibly a combination). In this case it is easy, because if they are packed upright, then their height is $20\text{cm}$, the same as the depth of the box. The width of each book is $15\text{cm}$, which goes twice into the width of the box; the thickness of each book is $6\text{cm}$, which goes six times into the length of the box. Thus $2\times 6=12$ upright books will fill the box completely, so no higher number is possible.