The Second Book of Mathematical Puzzles and Diversions

Tetraflexagons 31
gethe'r to make a flat square, but make sure that the two
inside flaps go in opposite directions (lle) . Open the square
as in drawing llf, then pull corner C down and to the left
to make the flat structure shown in drawing llg. Corner D
is now pushed to the left, behind the structure, creating the
flat rectangle of drawing llh. This rectangle opens to form
a cubical tube (lli) that is half the height of the original one.
You are now at the mid-point of your operations; exactly
half the tube has been reversed. Flatten the tube to make a
rectangle again (llj), but flatten it in the opposite way
from that shown in drawing llh. Starting as shown in
drawing Ilk, the previous operations are now "undone," so
to speak, by performing them in reverse. Result: a reversed
flexatube. At least two other completely different methods
of turning the flexatube inside out are known, both as de-
vious and difficult to discover as this one.
Recently Stone has been able to prove that a cylindrical
band of ang width can be turned inside out by a finite num-
ber of folds along straight lines, but the general method is
much too involved to describe here. The question arises : Can
a paper bag (that is, a rectangular tube closed on the bot-
tom) be turned inside out by a finite number of folds? This
is an unsolved problem. Apparently the answer is no, re-
gardless of the bag's proportions, though it probably would
be extremely difficult to find a satisfactory proof.