## Fall, 2018: Creativity Seminar in Math 22

One seminar of October 1, 2018 dealt with creativity. Here is the PDF. It especially mentions Fritz Zwicky, one of the giant astrophysicists of the 20th century who also thought a great deal about creativity.## July 5, 2018: Creativity through Computer algebra

The Blog entry Creativity through computer algebra illustrates how computer algebra projects can be creative. In the last part, I illustrate how one can come up with new questions.**July 2022:**the Medium.com blog page seems to be down. Here is the local version.

It mentions the square-graphs G(n) for which the vertices are the numbers 1 to n and where the edges are pairs (a,b) for which a+b is a square. One question which came out when experimenting with these graphs is that for n bigger or equal to 32, these graphs G(n) are Hamiltonian. The question was probed with the following code:

F[n_]:=Module[{e={}},Do[If[IntegerQ[Sqrt[x+y]], e=Append[e,x->y]],{x,n-1},{y,x+1,n}];e]; Puzzle[n_]:=FindHamiltonianCycle[UndirectedGraph[Graph[F[n]]]]; Do[If[Length[Puzzle[n]]>0,Print["G(",n,") is Hamiltonian"]],{n,1,152}]So, far we have checked that the graphs G(32), G(33),... G(151) are Hamiltonian. Here is a picture of the square graph G(151). So, far Mathematica gets stuck when checking whether G(152) is Hamiltonian. This illustrates that checking the Hamiltonian property is a hard problem. The actual paths look pretty random so that we also don't have a way to ``guess" the Hamiltonian path for larger n:

F[n_]:=Module[{e={}},Do[If[IntegerQ[Sqrt[x+y]],e=Append[e,x->y]],{x,n-1},{y,x+1,n}];e]; G[n_]:=Map[First,Flatten[First[FindHamiltonianCycle[UndirectedGraph[Graph[F[n]]]]]]]; ListPlot[G[151],Joined->True]I recently thought a bit about Hamiltonian graphs in in this project illustrated below. Unlike for general graphs, graphs which triangulate a manifold are Hamiltonian and the proof is constructive: