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5.F.4. OTHER CIRCUITS IN AND ON A CUBE
The number of Hamiltonian Circuits on the n-dimensional cube is the same as the number of Gray codes (see 7.M.3) and has been the subject of considerable research. I will not try to cover this in detail.
D. W. Crowe. The n dimensional cube and the Tower of Hanoi. AMM 63:1 (Jan 1956) 29 30.

E. N. Gilbert. Gray codes and paths on the n-cube. Bell System Technical Journal 37 (1958) 815-826. Shows there are 9 inequivalent circuits on the 4-cube and 1 on the n-cube for n = 1, 2, 3. The latter cases are sufficiently easy that they may have been known before this.

Allen F. Dreyer. US Patent 3,222,072 -- Block Puzzle. Filed: 11 Jun 1965; patented: 7 Dec 1965. 4pp + 2pp diagrams. 27 cubes on an elastic. The holes are straight or diagonal so that three consecutive cubes are either in a line or form a right angle. A solution is a Hamiltonian path through the 27 cells. Such puzzles were made in Germany and I was given one about 1980 (see Singmaster and Haubrich & Bordewijk below). Dreyer gives two forms.

Gardner. The binary Gray code. SA (Aug 1972) c= Knotted, chap. 2. Notes that the number of circuits on the n-cube, n > 4, is not known. SA (Apr 1973) reports that three (or four) groups had found the number of circuits on the 4-cube -- this material is included in the Addendum in Knotted, chap. 2, but none of the groups ever seem to have published their results elsewhere. Unfortunately, none of these found the number of inequivalent circuits since they failed to take all the equivalences into account -- e.g. for n = 1, 2, 3, 4, 5, their enumerations give: 2, 8, 96, 43008, 5 80189 28640 for the numbers of labelled circuits. Gardner's Addendum describes some further work including some statistical work which estimates the number on the 6-cube is about 2.4 x 10²⁵.

David Singmaster. A cubical path puzzle. Written in 1980 and submitted to JRM, but never published. For the 3 x 3 x 3 problem, the number, S, of straight through pieces (ignoring the ends) satisfies 2  S  11.

Mel A. Scott. Computer output, Jun 1986, 66pp. Determines there are 3599 circuits through the 3 x 3 x 3 cube such that the resulting string of 27 cubes can be made into a cube in just one way. But cf the next article which gives a different number??

Jacques Haubrich & Nanco Bordewijk. Cube chains. CFF 34 (Oct 1994) 12 15. Erratum, CFF 35 (Dec 1994) 29. Says Dreyer is the first known reference to the idea and that they were sold 'from about 1970' Reproduces the first page of diagrams from Dreyer's patent. Says his first version has a unique solution, but the second has 38 solutions. They have redone previous work and get new numbers. First, they consider all possible strings of 27 cubes with at most three in a line (i.e. with at most a single 'straight' piece between two 'bend' pieces and they find there are 98,515 of these. Only 11,487 of these can be folded into a 3 x 3 x 3 cube. Of these, 3654 can be folded up in only one way. The chain with the most solutions had 142 different solutions. They refer to Mel Scott's tables and indicate that the results correspond -- perhaps I miscounted Scott's solutions??