# CTK Insights

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20 May

### Probabilities in a Painted Cube

A wooden cube - after being painted all over - has been cut into $3\times 3\times 3$ smaller cubes. These were thoroughly mixed in a bag, from which one was produced and tossed. What is the probability that a painted side turned up?

On impulse, one would approach the problem in a more or less standard way. There are $8$ corner cubes with $3$ painted sides, $12$ mid-edge cube with $2$ painted sides, and $6$ face-central cubes with only $1$ side painted. The total probability is then

$\displaystyle\frac{8}{27}\cdot\frac{3}{6}+\frac{12}{27}\cdot\frac{2}{6}+\frac{6}{27}\cdot\frac{1}{6}=\frac{1}{3}.$

Now generalize: cut the cube into $n\times n\times n$, $n\gt 1$ smaller cubes and ask the same question. The problem is not awfully difficult but needs some figuring out. Following the foregoing pattern, we eventually arrive at

$\displaystyle\frac{8}{n^3}\cdot\frac{3}{6}+\frac{12\cdot (n-2)}{n^3}\cdot\frac{2}{6}+\frac{6\cdot (n-2)^2}{n^3}\cdot\frac{1}{6}$

This expression simplifies, as you can verify, to $\displaystyle\frac{1}{n},$ which, by the way, confirms the answer for $n=3.$

The above is a useful and not too difficult exercise, but there is a delightful shortcut that avoids most of the counting of cubes and their sides.

$n^3$ cubes have a total of $6\cdot n^3$ sides. Of these, $6\cdot n^2$ are painted. All sides have the same probability of turning up, therefore, a painted side will turn up with the probability $\displaystyle\frac{6n^2}{6n^3}=\frac{1}{n}.$

### References

1. R. Honsberger, Mathematical Delights, MAA, 2004, pp. 77-78

#### 6 Responses to “Probabilities in a Painted Cube”

1. 1
Joshua Zucker Says:

Even easier: As you look at the cube from any side at any location, you see one painted face out of the n painted faces aiming toward you along that column. So there's everywhere 1/n faces painted.

2. 2

Joshua, I am very much sorry but I do not understand that. Am I at any location on the cube looking along a column? If so, I probably see only a part of that column - this if making an effort not to see anything else, even a little sideways. In any event, what I see very much depends on my location. This is according to my interpretation of what you wrote. I do want to understand that.

3. 3
Joshua Zucker Says:

I'm imagining that I can see the "top" face of all n little cubelets in one column stacked up as I look down on the top. My use of "see" was probably too metaphorical as you certainly can't have transparent painted faces! Anyway in each column of n small cubes, and in each of the six directions of the faces, we'll have exactly one painted side, so 1/n of all the small faces are painted.

4. 4

Now I see! Standing on a tall (even semi)transparent column and looking all the way down may make one dizzy - depending on n of course. Thank you. That's a transparent way to think of the problem.

5. 5
Math Teachers at Play 63 | Math Jokes 4 Mathy Folks Says:

[...] Probabilities in a Painted Cube, Cut the Knot examines solutions to a problem about painting and cutting a larger cube into unit [...]

6. 6
Patty Tresca Says:

My tip: As you look at it from any side at any location, you see the painted face out of the n painted faces aiming toward you along that column. So there's everywhere 1/n faces painted.