Question 173
We have $n^2$ empty boxes, each of them having a square base. The height and the width of each box belong to $\{1, 2, \dots, n\}$ and every two boxes differ in at least one of these two dimensions. One box fits into another one if both its dimensions are smaller and at least one of them is smaller by at least $2$. In this way, we can form sequences of boxes (the first one in the second one, the second one in the third one, and so on). We put any such set of boxes on a separate shelf. How many shelves do we need to store all the boxes? (Peter Novotný)