2017 IMO Shortlist C6

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. (Russia)

令 $n>1$ 为整数。一个 $n \times n \times n$ 立方体由 $n^{3}$ 个单位立方体组成。每个单位立方体都涂有一种颜色。对于每个由 $n^{2}$ 单位立方体(三个可能方向中的任何一个)组成的 $n \times n \times 1$ 盒子，我们考虑该盒子中存在的颜色集(每种颜色仅列出一次)。这样，我们就得到了 $3 n$ 组颜色，根据方向分为三组。碰巧对于任何组中的每个集合，相同的集合都出现在其他两个组中。根据 $n$ 确定存在的最大可能颜色数。 (俄罗斯)