This is a full explanation for the question "100 Boxes, Four Cubes."

So we start with 100 cubes:

- 64 of them are arranged in a 4x4x4 cube
- 27 are arranged in a 3x3x3 cube
- 8 are arranged in a 2x2x2 cube
- 1 is by itself in a 1x1x1 cube

The outside faces of each of these arrangements are painted.

Now, think about each arrangement, and which sides of the cubes in each arrangement are painted.

Imagine the 4x4x4 cube as the following:

Blue = corner cube ==> 3-sides painted (8 total)

Green = edge cube ==> 2-sides painted (24 total)

Red = face cube ==> 1-side painted (24 total)

There are also 8 cubes in the interior of this figure with no exposed sides ==> 0-sides painted

And the 3x3x3 cube:

Corner cubes: 8 total

Edge cubes: 12 total

Face cubes: 6 total

interior cubes: 1 total

And the 2x2x2 cube:

Corner cubes: 8 total

And the single cube:

yellow = single-cube ==> 6-sides painted (1 total)

And now if we count each type of painted cube, we come up with the following table:

Now, we're going to arrange these 100 cubes as a 10x10 square. We must have paint on the tops and each exposed edge of the square. So something like this:

This means we must have 4 'corner' cubes at each corner of the square. Likewise, we need 32 edge cubes along the perimeter. And we'll need 100–4–32 = 64 face cubes in the middle.

We have 24 corner cubes to work with, so setting 4 in the corners is not a problem. This leaves us with 20 corner cubes left over

Likewise, we have 36 edge cubes, so we can set 32 along the perimeter, leaving us with 4 edges left over.

As for the 64 face cubes in the middle, we have 30 face cubes + 20 left over corners + 4 left over edges + 1 single cube = 55 faces in the middle.

To get the remaining 64–55=9 face cubes, we'll have to paint one side of the 9 blank interior cubes we have.

Thus, our answer is C) 9. :-)

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