The Rubik's cube is usually presented as "the puzzle with over a billion permutations". But how many permutations does the cube actually have?

The rubik's cube consists of 21 cubelets, which hold the cube together. However you cannot exchange edgepieces with centerpieces and cornerpieces etc. so that limits our number of permutations. To determine the number of possibilities to exchange certain pieces is calculated by using the factorial of the number of pieces.

For example: when I have 20 discs, and I wanna know in how many different orders I can lay down all my discs, I can first chose between 20 discs. Once I chose, I still have 19 discs to chose from left. And after that 18, etc. So there are 20*19*18*17*...*3*2*1 = 2.43*10^18 possible permutations. The long calculation is shortened by using !, because 20*19*...*2*1 = 20!.

So, we can permute our edges with 12! different permutations (because there are 12 edgepieces) and we can permute our cornerpieces with 8! different permutations (because there are 8 cornerpieces). We cannot exchange the centers with one another, because their location is fixed relative to the corepiece.

Our calculation is now 12! * 8! (* because you cannot choose whether to use corner- or edgepieces), but there is still one fault in there. We cannot exchange two cornerpieces with leaving the rest of the cube intact. That means we have to devide everything by 3: 12! * 8! / 3.

This looks like a nice calculation, but we forgot that we can also orient the pieces in different ways. Every corner can be placed correctly, and oriented in 3 different ways. So there are 3^8 possible ways to orient the corners. However, we cannot change the orientation of one cornerpiece without desturbing the rest of the cube. The orienatation of the cubelets always goes in pairs, so we need to devide our total by 2. Then, of course, we can also orient the edges in two different ways each, so there are 2^12 different orientations. However, we cannot flip an edge on its own, so we will need to devide it by an extra 2.

Let's see what we've got here. 12! * 8! * 2^12 * 3^8 / 2 * 2 * 3. When we calculate this, we come to 4.33*10^19 possible permutations, which is way more then "billions". In fact there are over 43 quintillion different permutations. And now consider that only one of those permutations is the right one. That's why turning the cube without thinking won't result in anything. Only structured solving methods will solve the cube.