Wolfram Cloud Document

Donald Knuth has posted results of an experiment “What a joy it is to learn [...] that my conjecture has a nice solution.”
https://www-cs-faculty.stanford.edu/~knuth/papers/claude-cycles.pdf

In a toroidal cubic lattice with odd edge-length n, can the edges be three-colored into Hamiltonian cycles? Knuth: “I had solved the problem for m = 3, and asked for a generalization as part of the answer to an exercise in [3]. My friend Filip Stappers rose to the challenge, and empirically discovered solutions for 4 ≤ m ≤ 16.”

They had solutions. Could they extend to infinity? Did a “nice” solution exist? A point figured out early was that no trivial solution existed. Eventually, they tossed the problem to LLM Claude. It managed to come up with a solution for odd values.

ClaudeAllCycles[m_Integer?Positive,start_:{0,0,0}]:=Module[{perm,step,cyc,vstr},perm[v_]:=With[{i=v[[1]],j=v[[2]],s=Mod[Total[v],m]},Which[s==0,If[j==m-1,{0,1,2},{2,1,0}],s==m-1,If[i>0,{1,2,0},{2,1,0}],True,If[i==m-1,{2,0,1},{1,0,2}]]];step[c_][v_]:=Switch[perm[v][[c+1]],0,{Mod[v[[1]]+1,m],v[[2]],v[[3]]},1,{v[[1]],Mod[v[[2]]+1,m],v[[3]]},2,{v[[1]],v[[2]],Mod[v[[3]]+1,m]}];cyc[c_]:=Most@NestList[step[c],start,m^3];vstr[v_]:=StringJoin[IntegerString[#,16]&/@v];(vstr/@cyc[#])&/@Range[0,2]]

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