Unit Cubes 1 – Counting Faces

Discrete Mathematics Level 2

Consider the outside of cubes made out of \( 1 \times 1 \times 1 \) unit blocks, with side lengths \( n \times n \times n \), for \( n \geq 2 \), which we have painted red on the outside.

The following table shows the number of unit cubes which are colored on a given number of faces.

Cube Size	3 faces red	2 faces red	1 face red
\( 2 \times 2 \times 2 \)	8	0	0
\( 3 \times 3 \times 3 \)	8	12	6
\(4 \times 4 \times 4 \)	?	?	?

What is the sum of the numbers missing from the row for the \(4 \times 4 \times 4 \) cube?

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