Probabilities in a Painted Cube
A wooden cube - after being painted all over - has been cut into 
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 
corner cubes with 
painted sides, 
mid-edge cube with 
painted sides, and 
face-central cubes with only 
side painted. The total probability is then
Now generalize: cut the cube into 
, 
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
This expression simplifies, as you can verify, to 
which, by the way, confirms the answer for 
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.
cubes have a total of 
sides. Of these, 
are painted. All sides have the same probability of turning up, therefore, a painted side will turn up with the probability 
References
- R. Honsberger, Mathematical Delights, MAA, 2004, pp. 77-78
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.
May 20th, 2013 at 6:35 pmJoshua, 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.
May 21st, 2013 at 8:55 amI'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.
May 23rd, 2013 at 4:57 pmNow 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.
May 25th, 2013 at 10:22 am[...] Probabilities in a Painted Cube, Cut the Knot examines solutions to a problem about painting and cutting a larger cube into unit [...]
June 14th, 2013 at 9:28 amMy 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.
January 13th, 2015 at 8:41 am