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Painted Cube Puzzle

A classic puzzle this week: The painted cube puzzle.

I’ve seen the problem printed in many locations, but I don’t know the original creator.

You have a 4” cube, and spray the outside with red paint. You cut this big red cube into individual 1” cubes (the freshly exposed faces do not have any red paint on them). All of these little cubes are placed into an urn, are thoroughly mixed, and one selected at random. This chosen cube is then rolled like a die. What is the probability that it lands with a red face up?

Solution

When the larger cube is cut up, 64 smaller cubes are created (4 x 4 x 4).

The 8 cubes that were in the middle of the larger cube have no red faces.
The 8 cubes that were at the corners have three red faces.
There are 24 cubes that were on the edges and have two red faces.
Finally there are 24 cubes that have just one painted face.

One of the 64 cubes is selected at random, and since we know how many of each ‘flavor’ of cubes there are, we can calculate the probability of each kind being selected. Then, since we know the number of painted faces on each kind of cube, we can find the overall probability.

P_r = (⁸/₆₄ × ⁰/₆) + (⁸/₆₄ × ³/₆) + (²⁴/₆₄ × ²/₆) + (²⁴/₆₄ × ¹/₆)

P_r = ⁰/₃₈₄ + ²⁴/₃₈₄ + ⁴⁸/₆₄ + ²⁴/₃₈₄

P_r = ⁹⁶/₃₈₄ = ¹/₄

There is a 25% that the cube will roll to red.

Alternate Solution

There's an alternative way to look at this problem and, to be honest, it's much simpler.

There are a total of 384 faces on all the mini-cubes (each of the 64 baby cubes has six faces = 64 × 6).

There are 96 faces that are painted red; the faces on the outside of the mother cube (6 × 16).

Each of these faces is equally as likely to be rolled = ⁹⁶/₃₈₄ = ¹/₄

The same answer. (Phew!)

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