Wikipedia:Reference desk/Archives/Mathematics/2017 April 20

= April 20 =

Rubik's Revenge all checkerboard
Is there a way to make all sides of a solved Rubik's Revenge cube be checkerboard? If you could, please show how with this style of notation: example r ² B ² U ² l U ² r ' U ² r U ² F ² r F ² l ' B ² r ². Many thanks! Anna Frodesiak (talk) 06:58, 20 April 2017 (UTC)
 * Not certain this will work, but you could try 180 degree rotations of alternate slices in each dimension i.e. 180 degree rotation of columns 1 and 3, rows 1 and 3 and layers 1 and 3. Gandalf61 (talk) 09:22, 20 April 2017 (UTC)
 * Hi Gandalf61. You are wise, but that only makes four out of six sides checkerboard. Anna Frodesiak (talk) 10:46, 20 April 2017 (UTC)


 * You could achieve your checkerboard if you could "swap front left middle with rear right middle". Achieving this swap requires some form of parity/entropy to be maintained and if that could be done as a matching "swap front right middle with rear left middle", then you have done the swaps for one of the three axes. Do it three times on the various axes. My guess is that it is doable, but I don't know how. -- SGBailey (talk) 14:02, 20 April 2017 (UTC)
 * Hi SGBailey. I just can't get my head around that. I'm so sorry. Anna Frodesiak (talk) 18:40, 20 April 2017 (UTC)
 * I hadn't realized you meant a 4*4*4 cube. -- SGBailey (talk) 06:45, 21 April 2017 (UTC)

An odd thing is that with a normal 3x3x3 Rubik's cube, one can easily checkerboard the whole thing. For some reason this doesn't work with Rubik's Revenge (4x4x4). It should because once the edges and center 4 are matched in colour, the 4x4x4 essentially becomes a 3x3x3. Mystifying! Anna Frodesiak (talk) 18:40, 20 April 2017 (UTC)

I watched through the following Youtube video https://www.youtube.com/watch?v=lnZWBq3C4Rs and given that someone who appears to be a very experienced cuber (or at least owns more types of these cubes that I'd ever seen before (up to 7x7x7, 2x3x3, 2x2x3,1x3x3 and a gear cube) and when he makes checkerboards for the even cubes it is only on four sides, I doubt it is possible. (Certainly not proof though)Naraht (talk) 19:11, 20 April 2017 (UTC)


 * It's not possible for any even x even x even, the reason being that the cubes aren't painted that way. So you can't even do by disassembling the cube and putting it back together.
 * Suppose you could. Then looking at the corner pattern from the top it might look like.

G  B O Y   W R R W   Y O   B   G
 * But now you have RWB in the upper right as well as the lower left, putting it in two places at once. --RDBury (talk) 22:39, 20 April 2017 (UTC)
 * PS. you could do it by disassembling four cubes and and reassembling them to create four different checkerboards. --RDBury (talk) 22:45, 20 April 2017 (UTC)


 * I think RDB is right, but it requires a more elaborate proof. The above assumes a particular way of constructing the checkerboard, but there are other ways to consider.  For example, in the above diagram one of the blue/green faces might have the two colors the other way around. Or the two colors checkerboarding a side might not be colors that were on opposite faces of the solved cube.  You have to prove that any such solution is impossible. --76.71.6.254 (talk) 23:14, 20 April 2017 (UTC)
 * You're right and I had a more detailed proof, but then I mistakenly thought it was more than needed. I think though, if you start with RWB in the lower left corner and try to fill in the remaining corners using a checkerboard pattern, the result would have to look like what's shown. The other possibilities you mention can only occur if you change the orientation of some of the corners. It is possible to disassemble a cube and a mirror image cube and put them together to form two checkerboards in three ways. --RDBury (talk) 01:53, 21 April 2017 (UTC)

I am still having trouble understanding, but the bottom line is what matters: I cannot be done. Fair enough. Many thanks to all for the thoughtful feedback. I am grateful. Best wishes and happy cubing! Anna Frodesiak (talk) 06:40, 21 April 2017 (UTC)