Consider door number 56, monkeys will visit it for every divisor it has. So 56 has 1 & 56, 2 & 28, 4 & 14, 7 & 8. So on pass 1 1st monkey will open the door, pass 2 2nd one will close it, pass 4 open, pass 7 close, pass 8 open, pass 14 close, pass 28 open, pass 56 close. For every pair of divisors the door will just end up back in its initial state. But there are cases in which the pair of divisor has same number for example door number 16. 16 has the divisors 1 & 16, 2 & 8, 4&4. But 4 is repeated because 16 is a perfect square, so you will only visit door number 16, on pass 1, 2, 4, 8 and 16… leaving it open at the end. So only perfect square doors will be open at the end.
Different means that the cubes can not be rotated so that they look the same. This is important! If you give me two cubes and i can rotate them so that they appear identical in color, they are the same cube.See Solution : Cube puzzle
Let X be the number of “different” cubes (using the same definition as in the problem). Let Y be the number of ways you can “align” a given cube in space such that one face is pointed north, one is east, one is south, one is west, one is up, and one is down. (We’re on the equator.) Then the total number of possibilities is X * Y. Each of these possibilities “looks” different, because if you could take a cube painted one way, and align it a certain way to make it look the same as a differently painted cube aligned a certain way, then those would not really be different cubes. Also note that if you start with an aligned cube and paint it however you want, you will always arrive at one of those X * Y possibilities.
How many ways can you paint a cube that is already “aligned” (as defined above)? You have six options for the north side, five options for the east side, etc. So the total number is 6! (that’s six factorial, or 6 * 5 * 4 * 3 * 2 * 1). Note that each way you do it makes the cube “look” different (in the same way the word is used above). So 6! = X * Y.
How many ways can you align a given cube? Choose one face, and point it north; you have six options here. Now choose one to point east. There are only four sides that can point east, because the side opposite the one you chose to point north is already pointing south. There are no further options for alignment, so the total number of ways you can align the cube is 6 * 4.
Remember, Y is defined as the number of ways you can align the cube, so Y = 6 * 4. This gives us 6! = X * 6 * 4, so X = 5 * 3 * 2 * 1 = 30.
Reference : http://www.techinterview.org/Solutions/fog0000000128.html