1999 CMO 第 1 题

A $4\times4\times4$ cube is composed of $64$ unit cubes. The faces of $16$ unit cubes are to be coloured red. A colouring is called interesting if there is exactly $1$ red unit cube in every $1\times1\times 4$ rectangular box composed of $4$ unit cubes. Determine the number of interesting colourings.

一个 $4\times4\times4$ 立方体由 $64$ 单位立方体组成。 16 美元单位立方体的面应涂成红色。如果每个由 $4$ 单位立方体组成的 $1\times1\times 4$ 矩形框中恰好有 $1$ 红色单位立方体，则该着色被称为有趣。确定有趣的颜色的数量。