Why Are There 43,252,003,274,489,856,000 Rubik's Cube Combinations?

why are there 43 quintillion 252 quadrillion I'm not gonna say the whole thing am i why are there over 43 quintillion possible positions on the Rubik's Cube instead of thinking about combinations of turns we can do on the cube we have to think about this cube as what it really is the Rubik's cubes six center pieces are all connected by a core which means they cannot move relative to each other so we can consider these as fixed in place and everything else moves around it to count how many possible positions there are on a cube it helps to just think about it as putting pieces in one by one there are 12 edge pieces and eight corner pieces now the difference between these is an edge pieces two colors and a corner piece has three colors edge pieces can go between two Center pieces such as this and a corner piece would go between three Center pieces such as this since corner and edge pieces belong in completely different spots they can't interfere with each other so for now we'll consider them as completely separate let's see what happens when I assemble just with edges first so for the very first edge I can put this in any position between two centers that means that right now I have 12 possible choices and I can just pick one of those spots now notice that I could have put it in here with green on the side or I could have put it in with orange on the side this is something I'll talk about later so we go on to the next edge and now we look at how many positions I have well I can't put it in the same spot as the previous one so at this point I can choose between eleven possibilities and I will just put it right here for my next edge I have ten possible places I can put it because two out of the twelve are already taken these were the number of options I had at any point and when you put them all together what you do is you multiply all of them if you've learned probability in school before it may seem obvious why you should multiply these together but to visualize why you would want to multiply think about it as a branching path through time from the start I had 12 possible choices and down each of those paths I had 11 choices I could make after that so just for the first two pieces the total number of paths I could have gone down where 12 times 11 it's too much work to draw the rest of this but you get the idea the total number of branches we'll be gigantic and we can simplify this all down to 12 factorial which just means a multiply 12 with every whole number of smaller than it for the corner pieces it's the exact same idea I have 8 places that I can put the first corner piece end so I'll just pick one of those spots and for the next corner piece one spot is taken so I only have 7 spots remaining I can put it in and again for the last corner piece I only have one spot it can go into okay I didn't think this through you're not supposed to put a cube back together like this for the corner pieces we have 8 times 7 times 6 all the way down to 1 and this is 8 factorial when you multiply all this together you get over 19 trillion different ways that you can put the pieces in the cube but I haven't talked about what if we flipped these pieces in different ways this edge for example could be in the same spot but flipped in the other way so you can think about it this way when I put the first edge in its spot I had 12 choices of where to put it but at each of those 12 spots I can choose to have it flipped one way or the other way which is another two options so there are actually twice as many options of what to do with the first edge and then twice as many for the next and the next and so on so we multiply all those together or you can also say that this is two to the power of 12 and then four corners this for example yellow on top or on the right or on the left so with each corner I multiplied by three there are eight corners so with the same reasoning as edges we have 3 to the power of 8 that's about it for how pieces can be arranged on the cube and if we multiply all these together we get 519 quintillion this is the correct answer for what I was calculating which is how many ways can you put the Rubik's Cube back together so this is just the cube reassembled and let's see what happens if I try to solve it okay twisted corner and flipped edge alright and this is not a possible PLL case so just to show you what I mean here are two swapped edges and that's not supposed to be possible in that solve I ran into three things that were impossible to solve just by turning the cube normally first thing is it's impossible to have one corner twisted and no other corners twisted the same thing applies for edges you cannot just have one edge flipped and the last thing is you cannot just have two pieces swapped so going back to the numbers let's change this to reflect what I just talked about in terms of how I put the corners in like how they're twisted I can put the first seven corners in however I like but the last one depends on how the first seven are twisted so remember the numbers here are just for assembling the cube anyway I want but if I want to assemble the cube to be solvable then the last three doesn't belong there and it should be a 1 because for the last corner I only have one option of how I put it in if I put it in wrong then the cube is not solvable and the same idea applies to edges the last one has to be flipped just the right way so that I don't end up with one flipped edge at the end so again I only have one option here and the last change we make here is since I've put in all the edges followed by all the corners I can actually put the edges wherever I like then I could put the corners wherever I like down to the last two for the last two corners if I put them in the wrong positions then I end up with an extra swap on the cube which is not solvable but if I put them in the right way then it's fine so for the last two corners there's only one option that actually keeps the cube solvable so instead of having two there should only be one if you work this out again you get the right answer which is 43 quintillion if you're a speed cuber you should already know that you can't have single corner twists you can't have a single edge flip and you cannot just swap two pieces but maybe you don't know why and if you notice B a cube or you probably never heard of this before so you're also wondering why that's the case the reason has to do with what one individual turn can do to the pieces now for why we have those very specific rules about twisting one or flipping one or swapping two pieces then you can watch this video where I go to in depth into the reason so that's it for 3x3 but I have a challenge for you guys how many positions are there on a 2x2 now you might think this is easy just grab the part on my 3x3 calculation that is just four corners and you're done but nope that's not the right answer and it's trickier than that so see if you can get the number and explained to me in the comments of why that is thanks for watching and I'll see you guys next time [Music] you

Channel: J Perm

Views: 1,578,099

Keywords: rubik's cube, tutorial, advanced, j perm, jperm, tips, tricks, speedcube, cubing, best, 3x3, budget, fast, faster, easy, easiest, intuitive, rubix cube, permutations, positions, probability, math

Id: z2-d0x_qxSM

Length: 6min 54sec (414 seconds)

Published: Tue Jun 23 2020