OlympiadBench / 1818 / Let n>1 be an integer. An n×n×n cube is composed of n³…

Let n>1 be an integer. An n×n×n cube is composed of n³ unit cubes. Each unit cube is painted with one color. For each n×n×1 box consisting of n² unit cubes (of any of the three possible orientations), we consider the set of the colors present in that box (each color is listed only once). This way, we get 3n sets of colors, split into three groups according to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of n, the maximal possible number of colors that are present.

Let $n>1$ be an integer. An $n \times n \times n$ cube is composed of $n^{3}$ unit cubes. Each unit cube is painted with one color. For each $n \times n \times 1$ box consisting of $n^{2}$ unit cubes (of any of the three possible orientations), we consider the set of the colors present in that box (each color is listed only once). This way, we get $3 n$ sets of colors, split into three groups according to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colors that are present.