Mock AIME 2 2006-2007 Problems/Problem 15

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A $\displaystyle 4\times4\times4$ cube is composed of $\displaystyle 64$ unit cubes. The faces of $\displaystyle 16$ unit cubes are colored red. An arrangement of the cubes is $\mathfrak{Intriguing}$ if there is exactly $\displaystyle 1$ red unit cube in every $\displaystyle 1\times1\times4$ rectangular box composed of $\displaystyle 4$ unit cubes. Determine the number of $\mathfrak{Intriguing}$ colorings.



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