A 4"x4"x4" cube is painted and then cut into sixty-four 1"x1"x1" cubes. A unit cube is then randomly selected and rolled. What is the probability that the top face of the rolled cube is painted? Express your answer as a common fraction.
Now, originally, I tohught the answer was 96/384, but I realized it was wrong since some sides (the corner cubes) have more than one painted side.
This is actually a cool problem. There are patterns to the number of painted faces on a nXn cube.
For a 3X3X3 cube, there are 8 cubes with 3 painted faces. Any cube will always have 8 cubes with 3 painted faces. Those are the corners.
There are 12 cubes with 2 faces painted. 6 cubes with 1 face painted and 1 with no faces painted(the one in the dead center).
That is 54 painted faces with 162 possible sides. 1/3 probability.
Now, for the 4X4X4 cube:
There are 8 with 3 faces painted. 24 with 2 faces painted. 24 with 1 face painted. 8 with 0 faces painted. That's 96 painted faces out of 384 possible sides. 96/384=1/4 probability. You are correct.
Now for a 5X5X5:
There are 8 with 3 faces painted. 36 with 2 painted. 54 with 1 painted and 27 with 0 painted
That's 150 painted sides out of 750 sides. 150/750=1/5.
See the pattern?. The probability of rolling a painted side is 1/n.
If it's a 10X10X10 cube, it's 1/10. Cool, huh?.
Therefore, we have a formula for the number of 2-painted sides: 12(n-2)
The number of 3 painted sides is always 8.
The number of 1-painted sides:
The number of no painted sides:
The number of cubes is then