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Problem 599

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Distinct Colourings of a Rubik’s Cube

The well-known Rubik’s Cube puzzle has many fascinating mathematical properties. The 2×2×2 variant has 8 cubelets with a total of 24 visible faces, each with a coloured sticker. Successively turning faces will rearrange the cubelets, although not all arrangements of cubelets are reachable without dismantling the puzzle.

Suppose that we wish to apply new stickers to a 2×2×2 Rubik’s cube in a non-standard colouring. Specifically, we have $n$ different colours available (with an unlimited supply of stickers of each colour), and we place one sticker on each of the 24 faces in any arrangement that we please. We are not required to use all the colours, and if desired the same colour may appear in more than one face of a single cubelet.

We say that two such colourings $c_1,c_2$ are essentially distinct if a cube coloured according to $c_1$ cannot be made to match a cube coloured according to $c_2$ by performing mechanically possible Rubik’s Cube moves.

For example, with two colours available, there are 183 essentially distinct colourings.

How many essentially distinct colourings are there with 10 different colours available?

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魔方上色

著名的鲁比克立方体（魔方）有着许多惊艳的数学性质。2×2×2的魔方由8个小立方体构成，共有24个可视的面，每个面上有一张彩色贴纸。合法地旋转操作可以重新排列这些小立方体，但有些小立方体的排列无法在不破坏魔方的前提下达成。

假设现在我们打算给2×2×2魔方换一套贴纸颜色。具体来说，我们有$n$种不同的可选颜色（每种颜色都有无限量的供应），然后在24个面的每一个上任选一种颜色的贴纸贴上去。我们并不要求使用全部的颜色，而且在同一个小立方体的不同面上可以有同样的颜色多次出现。

我们称两种上色方案$c_1,c_2$是完全不同的，如果按照$c_1$上色的魔方不能通过机械上可行的魔方操作转化为按照$c_2$上色的魔方。

例如，如果只有两种可选的颜色，那么一共有183种完全不同的上色方案。

如果有10种不同的可选颜色，一共有多少种完全不同的上色方案？

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