Hello, Brent!

It's simple arithmetic . . .

6. A 2-by-4-by-8 rectangular solid is painted red.

It is cut into unit cubes and reassembled into a 4-by-4-by-4 cube.

If the entire surface of this cube is red,

how many painted unit-cube faces are hidden in the interior cube? Code:

8
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/ /|
4 / / | 2
/ / |
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2 | | / 4
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8

The total surface area is:

. . $\displaystyle \begin{array}{cccccc}\text{top/bottom:} & 2\times(8\times4) &=&64 \\

\text{left/right:} & 2\times(4\times2) &=& 16 \\

\text{front/back:} & 2\times(8\times2) &=&32 \\\end{array}\quad\Rightarrow\quad 112$

There are 112 red faces on the sixty-four unit cubes.

Code:

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/ /|
4/ / |
/ / | 4
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| | *
4 | | /
| | /4
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4

The total surface area of the cube is: .$\displaystyle 6 \times (4\times4) \:=\:96$

There are 96 red faces on the outside of the cube.

Therefore, there are: .$\displaystyle 112 - 96 \:=\:\boxed{16}$ red faces hidden inside.