A Bit of Preliminaries
Magic squares, but not Magic Flexagons, have been of persistent interest for many a century. Flexagons have only been around since 1939, and have not until now been found capable of joining the ranks of the magic squares, magic hexagons, and magic stars. The many variations of flexagons have been studied concentrating mostly on their artistic and manipulation idiosyncrasies.

Furthermore, it was believed until now that there was only one version of the trihexaflexagon. Or, at least, references to only one version has ever been published with absolutely no hint that another version had been ever studied. Along with my discovery of the other configuration, I found that trihexaflexagons had 'magic' in addition to seemingly endless new applications.

Now To My 'Magic' Flexagon!
This magic flexagon, similar in concept to a magic square, was found by numbering the vertices (corners) of the triangles on the two edges of the paper band consecutively 1 through nine until the inevitable return to the starting point (3 sets of 9 for nine triangles having their vertices labeled on one side). The reverse side vertices were given the same number appearing on the other side of the band. I added the symbol,' to that other side number in order to represent the idea that the Möbius band has no thickness along with its having only one side (see ' numbered strip)).

It was only a few days later that I realized what happened when I numbered the vertices of this one-sided strip with 18 triangles. I numbered the vertices 1-9 in one color, then, 1-9 in a second color followed by 1-9 in a third color. After repeating this sequence a second time in order to complete numbering the 54 vertices of the 18 triangles, the number was always the same, and, it was the same color on each 'side' of the vertice. Interesting!

Flexagons are made using Möbius Bands. I find that my 'Magic' Flexagons are a great help when explaining to people that a Möbius Band has only one side and one edge. The repeat of the same color number on opposite sides of the band, plus, the same numbered triangles on opposite sides of the band helps in explaining the one side and one edge attributes of the Möbius Band. Furthermore, the similarity to a 3X3 magic square which adds to 15 in each row, column, and diagonal is interestingly present in this trihexaflexagon. Each triangle adds to fifteen. Each face adds to 90 and each of the three pairs add to 180.


color numbered strip and colored numbered faces.
Apparently, either the repetition of any 9 consecutive numbers or multiples of 9 consecutive numbers result in each triangle's vertices summing to an identical number. Now what would be the mathematical proof or proofs related to this adventure? Could it be proven that there is only one possible set of solutions? Do all the triangles have to have identical sums and can that be proven?

Hmmmmm. This next is of course pure fantasy, but I can't resist. Could it possibly be that Fermat also trimmed the margins of books where he had so often written notes in order to discard what he felt were errors? What if he had played with these strips of paper and eventually found a flexagon? Is there a yet to be found 'Fermat'agon lurking somewhere? I wonder if Fermat had a comparable 'magic' flexagon's last theorem hidden somewhere in an obscure book margin?