So any number of flipped edges can be shimmied down to a single edge, which winds up either flipped or not, for two possibilities. There’s an algorithm that flips, in place, each of two different edges, but no algorithm can flip a single edge in place. Overall, the corners’ orientations can go one of three ways. If the other one happens to get fixed in the process, we got lucky and now we’re back to a solvable cube. Now that two corners are twisted, we can apply the algorithm that twists two corners, until at least one is fixed. However, if you repeat that process, and twist one more corner, it doesn’t add a second factor of 3. So if you grab a normal Rubik’s Cube, pry out a single corner, and replace it twisted, it becomes impossible to solve, and you’ll have moved from the top-left corner of our chart to one of the spots right below it. The factor of 3 comes down to this: There’s an algorithm that twists each of two different corners, but there’s no algorithm that twists a single corner (while leaving everything else unmoved). We need to grapple with a factor of 3, and two factors of 2. That 12 comes together from three factors getting multiplied: 12 = 3*2*2. The limitations to the algorithms are the key to that number 12. The sought-after algorithms are those that move just a few of the cubies while leaving the rest untouched. A sequence of moves is often referred to as an “algorithm” by cube enthusiasts. There’s a connection here with what can and can’t be done by moving the cube’s faces. And there’s no 13th arrangement that can’t transform into one of those 12. So the 12 cubes in the photo above can’t be transformed into one another.
Rubik cube flip two corners plus#
Finally, column 4 has one edge flipped plus two edges swapped. Rows 2 and 3 have one corner rotated in place. We’ve arranged these to splay out the different factors leading to 12. A cube that’s been broken and reassembled with its cubies randomly scrambled will have equal chances of being solvable to one of the following representatives. Want to understand why that’s 1 in 12? Well, there’s a nice visual way to get a sense of it. If you break it apart and reassemble the cubies randomly, there’s actually only a 1 in 12 chance that it’s solvable. If you’re practicing and you want to scramble a solved cube, you have to keep the cube intact and scramble it up manually. This is a trap that has caught many novice cubers. Now, is it always possible to solve this jumbled cube, without breaking it apart? You get what looks like a normal scrambled cube, and so far we’ve counted every way you could do this, (3 88!)(2 1212!). Suppose you break open a Rubik’s Cube, remove each cubie, and then put all the cubies back in random slots (with corner cubies only fitting in the corners, and edge cubies only in the edges).
Here’s a thought experiment (which perhaps you’ve done for real!) to illustrate: It relates to a fact about Rubik’s Cube that is often felt, but not always understood. What’s left of the formula (3 88!)(2 1212!)/12 is that division by 12. Then there are 12 locations, so 12! is the number of ways they can go to those spots. Edges only have two orientations, so the 12 of them have a total of 2 12 mixes of orientations. The next chunk, (2 1212!), is the same idea, now for the edges. The 3 8 is their orientations, while the 8! is their locations. Thus the first chunk, (3 88!), counts every way the corner cubies can fit into the cube. That yields the calculation 8*7*6*5*4*3*2*1, which is 8!, or “eight factorial.” The second corner cubie is left with seven options, the next left with six, and so on, down to the last corner cubie, which must go in the last corner slot. There are eight corner slots, so the first corner cubie has eight options.
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