The Number of Flexagons

It might be said that the most interesting mathematical problems concerning flexagons arise when one begins to count them. Certainly Oakley and Wisner's treatment of the triflexagons shows both that these combinatorial problems are non-trivial and that their solutions may throw light on the flexagon problem in general.

However, aside from the results which Oakley and Wisner have obtained, it appears that little is known in this direction, and the present authors cannot at this time produce many answers. Still, it seems worthwhile to at least formulate some of the remaining problems, in the hope that this will encourage solutions.

By analogy with the problem of counting triflexagons of a given order, we might ask how many G-flexagons there are of order $N$. This question still allows ambiguity: are we to allow incomplete cycles or not?. If we do, we are essentially counting the total number of different flexagons of cycle less than or equal to $G$ or of mixed cycles less than or equal to $G$. This seems superficially related to the problem of the total number of flexagons, with mixed or non-mixed cycles, with order $N$; i.e., the number of dissections of a polygon into polygons by non-crossing segments joining the vertices. Then again, we might not include incomplete cycles at all, thus seemingly simplifying the question considerably.

In any of these cases, we have not considered face degrees at all. We might ask, for instance, how many different 1-flex cycles one could have about a single cycle of a G-flexagon, as shown in figure 16.1 for $G<7$. Or, if partially tubulating flexagons are not considered, how many different possible arrangements in all are there per G-cycle?. Then again, we may decide not to allow zero-faces in each of these cases. All of these problems are extended by asking how many different G-flexagons can be made with K-gons, for $K > G$.

There is always the problems of counting the flexagons whose maps possess some special property, such as rotational or mirror-image symmetry or being made up entirely of some pat structural unit, as with the regular triflexagons. The number of self-dual flexagons is not known even for $G=3$, and no fruitful work at all has been done on the number of flexagons derivable from a given sign sequence, nor on the number of distinct sign sequences of a given order.

There are one or two easier questions whose answers can be supplied readily, to end on a promising note. The total number of faces in the G-flexagon is twice the number of paths. If cut-offs are not included in the count, there will be $2 \frac {NG-N-G} {G-2}$ faces; if cut-offs are included, this becomes (G+1)(N-2)+2. These figures hold only when applied to complete flexagons.

In order to find the number of map paths touching a given side, without drawing the map, proceed as for triflexagons, but reducing sequences of $G-1$ consecutive numbers instead of mere pairs of consecutive numbers. In this way a minimum number $M$ of different leaves are eventually left. The number of paths touching the given side is then $\frac {M-2} {G-2}+1$; note that this is just $M-1$ for $G-3$ and $1/2 M$ for $G-4$. If cut-offs are to be counted among the map paths touching the face, then the number of such paths is simply $M-1$.