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Arthur Stone's Flexatube

Folding Puzzles

by Serhiy Grabarchuk


 
Arthur Stone's Flexatube can be turned inside out.
 
Printable Puzzle
The Flexatube's layout is a four-square strip.
It is a great and unique puzzle discovered in 1939. Martin Gardner in his Second Scientific American Book of Mathematical Puzzles and Diversions (1961),1 writes that Arthur Stone discovered the Flexatube somewhat accidentally working on a flexagon, and was very surprised with this discovery.

The one hardly can imagine a puzzle which is so simple to make, and so interesting and hard to solve as the Flexatube is. Actually, this is a simple paper tube - a paper cube without top and bottom faces. It can be folded along its edges and its faces' diagonals; all these lines should be previously pre-creased to have smooth folding. The object of the puzzle is to turn the tube inside out, folding it only along the lines between the triangles, and without any other bends, folds or other deformations of paper.

There are several different correct solutions to the Flexatube found in different years by Arthur Stone, T. S. Ransom, Hugo Steinhaus,
David Mitchell,2 Stefan Louw, and myself,3 but anybody who managed to find even one solution to it can say how interesting and hard at the same time the solving process is.

A very surprising fact is that the very first solution of the Flexatube has an intermediate step (exactly in the midway) where the Flexatube changes into... its smaller copy - a small cubical tube! Great tube-reducing trick!

Because of its external and internal elegancy and mathematical harmony the Flexatube can be considered as one of the best puzzles ever.

To solve the Flexatube you can easy make it, using provided image of the unfolded puzzle. First, print the Flexatube layout. For this, click the image at left; it's marked with our special "printable puzzle" pictogram. You will go to a new window with the Flexatube layout; then you can print and cut it out. Then carefully pre-crease and pre-fold the puzzle strip along all dotted lines. And, finally, paste together two opposite ends of the strip; the gray flap is the place for glue.

Note that you can print the puzzle for your own use only, and not for any kind of commercial profit.
References
     1) Martin Gardner, The Second Scientific American Book of Mathematical Puzzles and Diversions, Simon & Schuster, New York, 1961; republished in 1987 by University of Chicago Press, Chicago.
(Chapter 2, "Tetraflexagons," pp. 24-31, describes Arthur Stone's Flexatube and one solution to it.)
     2)
David Mitchell, Paperfolding Puzzles, Water Trade, Kendal, 1998.
(It is a small but fascinating book with plenty of folding puzzles! It presents several different solutions to the Flexatube.)
     3)
Serhiy Grabarchuk, Age of Puzzles: Paperfoldings, Serhiy Grabarchuk Puzzles, 2005. (It shows my original solution of 1978 to the Flexatube.)
 
Last Updated: September 28, 2009
Posted: May 21, 2007
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