Date: 24 May 1982 10:56-EDT From: David C. Plummer Subject: Ideal and the C^4 To: RP at MIT-MC cc: CUBE-LOVERS at MIT-MC I have done a complete search on paper for a complete checkboard. It cannot be done for any cobe of even side. The restrictions are in the corners. I would like somebody to double check this though. The standard problems to run into are: You need two of one type of corner, or You have to rotate exactly one corner, which is impossible. There are a couple other amazing things I found. As it is known, any two edges can be exchanged (or appear to be exchaged because of center arbitrariness). It is ALSO possible to (appear to) exchange two corners, for about the same reason. This is impossible on the 3^3 becuase it requires the exchange of two edges. But in the 4^3 there are two cubies per 3^3 edge. Therefore, we just do a double exchange, which does not violate any parity arguments. Combining these two moves, you can flip a 1x1x4 edge. Happy revenge.