This is a relat ively simple problem that can be fiendishly difficult unless you have a good approach to solving it and a solid understanding of how to count. We will present two different strategies here. Strategy 1: This problem seems difficult, because you need to figure out how many distinct orientations the cube has relative to its other sides. Given that you can rotate the cube in an unlimited number of ways, it is very difficult to keep track of what is going on – unless you have a system. Big hint: In order to analyze how mult iple things behave or compare or are arranged relative to each other, the first thing one should do is pick a reference point and fix it. Here is a simple example. Let’s say you have a round table with four seat positions and you want to know how many distinct ways you can orient 4 people around it relative to each other (i.e., any two orientations where all 4 people have the same person to their left and to their right are considered equivalent). Let’s pick person A as our reference po int and anchor her to the North position. Think about this next statement and convince yourself that it is true: By choosing A as a fixed reference, all distinct arrangements of the other 3 people relative to A will constitute the complete set of distinct arrangements of all 4 people relative to each other. Hence, fixing the location of one person makes it significantly easier to keep track of what is going on. Given A is fixed at the North, the 3 other people can be arranged in the 3 remaining seats in 3! = 6 ways, so there are 6 dist inct orientations of 4 people sitt ing around a circular table. Using the same principle, we can conclude that, in general, if there are N people in a circular arrangement, after fixing one person at a reference point, we have (N-1)! distinct arrangements relative to each other. Now let’s solve the problem. Assume the six sides are: Top (or T), Bottom (or B), N, S, E, and W, and the six colors are designated 1, 2, 3, 4, 5, and 6. Following the first strategy, let’s pick color #1 and fix it on the Top side of the cube. If #1 is at the Top position, then one of the other 5 colors must be at the Bottom position and each of those colors would represent a distinct set of arrangements. Hence, since there are exactly 5 possible choices for the color of the Bottom side, the number of unique arrangements relative to #1 in the Top position is a multiple of 5. For each of the 5 colors paired with #1, we need to arrange the other 4 colors in the N, S, E, and W positions in distinct arrangements. Well, this is exactly like arranging 4 people around a circular table, and we have already determined that there are (n-1)! or 3! ways to do that. Hence, the number of distinct patterns of painting the cube is simply 5 x 3! = 30. 
Strategy 2: There is another way to solve this kind of problem. Given one dist inct arrangement or pattern, you can try to determine how many equivalent ways there are to represent that one particular arrangement or pattern within the set of total permutations, then divide the total number of permutations by that number to get the number of distinct arrangements. This is best illustrated by example so let’s go back to the 4 people arranged around a circular table. Assume A is in the North posit ion, then going clockwise we get B, then C, then D. Rotate the table 1/4 turn clockwise. Now we have a different arrangement where D is at the North position, followed clockwise by A, then B, then C. BUT, this is merely a rotation of the distinct relative position of the 4 people (i.e., everyone still has the same person to his right and to his left) so they are actually the same arrangement. We can quickly conclude that there are 4 equivalent or non-dist inct arrangements for every dist inct relat ive posit ioning of the 4 people. We can arrange 4 people in a total of 4! = 24 ways. However, each DISTINCT arrangement has 4 equivalents, so in order to find the number of distinct arrangements, we need to divide 4! by 4, which yields 3! or 6 distinct ways to arrange 4 people around a circular table, the same result we got using the “fixed reference” method in Strategy 1. Generalizing, if there are N! ways to arrange N people around a table, each dist inct relat ive rotation can be represented in N ways (each 1/Nth rotation around the table) so the number of dist inct arrangements is N!/N = (N-1)! Now let’s use Strategy #2. Consider a cube that is already painted in a particular way. Imagine putting the cube on the table, with color #1 on the top side. Note, that by rotating the cube, we have 4 different orientations of this particular cube given color #1 is on top. Using symmetry, we can repeat this analysis when #1 is facing any of the other 5 directions. Hence, for each of the six directions that the side painted with #1 can face, there are 4 ways to orient the cube. Consequently, there are 6 x 4 = 24 total orientations of any one cube painted in a particular manner. Since there are 6 sides and 6 colors, there are 6! or 720 ways to color the six sides each with one color. However, we have just calculated that each DISTINCT pattern has 24 equivalent orientations, so 720 must be divided by 24 to get the number of distinct patterns. This yields 720/24 = 30, confirming the answer found using Strategy #1.