The Second Book of Mathematical Puzzles and Diversions

Tetraflexagons 29
the lines were originally creased. The strip now looks like
10c. Fold on the three lines marked with arrows to form a
square flexagon. Overlap the ends so that all the "2" squares
are uppermost (10d). Attach a piece of transparent tape to
the edge of the square at upper left, then bend it back to
overlap the edge of a "1" square on the opposite side.
The hexa-tetraflexagon can now be flexed along both ver-
tical and horizontal axes to expose all six of its faces.
Larger square strips will yield flexagons whose number of
faces increases by fours : 10, 14, 18, 22 and so on. For tetra-
flexagons of different orders, strips of other shapes must
be used.
It was while Stone was working on right-triangle forms
of flexagons ("for which, perhaps mercifully," he writes in
a letter, "we invented no name") that he hit upon a most
remarkable puzzle - the tetraflexatube. He had constructed
a flat, square-shaped flexagon, which to his surprise opened
into a tube. Further experimentation revealed that the tube
could be turned completely inside out by a complicated
series of flexes along the boundaries of the right triangles.
The flexatube is made from a strip of four squares [see
Fig. 111, each of which is ruled into four right triangles.
Crease back and forth along all the lines, then tape the ends
together to form the cubical tube. The puzzle is to turn the
tube inside out by folding only on the creased lines. A more
durable version can be made by gluing 16 triangles of card-
board or thin metal onto cloth tape, allowing space between
the triangles for flexing. It is useful to color only one side of
the triangles, so that you can see at all times just what sort
of progress you are making toward reversing the tube.
One method of solving this fascinating puzzle is illus-
trated in drawings llb through Ilk. Push the two A corners
together, flattening the cube to the square flexagon of draw-
ing llc. Fold this forward along the axis BB to form the
triangle of drawing lld. Now push the two B corners to-