# «Contents Ü Foreword Elwyn Berlekamp and Tom Rodgers ½ I Personal Magic ¿ Martin Gardner: A “Documentary” Dana Richards ½¿ Ambrose, Gardner, ...»

About a year or so ago, I discovered a way to make a ﬂexagon without removing any portion of the square. (The goal was inspired by my desire to make a ﬂexagon from paper money with minimal damage.) It is made from a square with two slits intersecting the center in the shape of a plus sign (Figure 3).

For best results, use the Alternating Way fold. Flexing is continuous, and four faces can be shown: checkerboard, solid white, solid black, checkerboard. The Same Way Fold can be used also, but with poor results. Four faces can be shown, but they are not continuous. Indeed, both the third and fourth are derived from the ﬁrst face. All faces are of the identical solid color. Using the 3-D Tricky Fold gives the same results as the Alternating Way Fold.

More recently, I have worked out some variations on the theme of removing no paper. These follow.

Square Slash-Slit

It must be a slash along the diagonal, however (Figure 4). The model will show four faces. There are two versions, the second allowing some presentational by-play. Use of paper money is recommended.

Using the Alternating Way Fold, ﬂexing is not continuous, and two of the sides are just a little tricky to ﬁnd. The designs are these: checkerboard;

checkerboard; solid white; triangle in each corner, leaving a white square in the center.

Using the Same Way Fold, ﬂexing is not possible without an additional move. The obvious way is to open up each side of the slit, and then move the two little ﬂaps inside the slit to a position outside it. Then ﬂexing is possible. Or, once you understand what is required, you can tuck the ﬂaps properly as you make the valley folds. However, and best of all, the move can be done quite secretly in the course of doing the ﬁrst ﬂex. Being sure you have the right axis on which to fold the model in half; do the folding, but pull gently as you do it, and the little ﬂaps will be adjusted automatically and properly, to be seen only at the very end of the ﬂexing. Obviously, you can create some mischief by folding, ﬂexing and unfolding, then challenging the knowledgeable paperfolder to duplicate the action. The ﬂexing is continuous. The faces are these: solid white; two black (and six white) triangles opposite each other; two white triangles opposite each other, arranged differently from the previous two; a second solid white.

Square X-Slit These models are made from squares with two intersecting slits making an X (Figure 5). Many versions of these are possible. Here are four of them.

All have six faces.

Use of the Same Way Fold shows four faces continuously, and the other two easily, in the following manner: face 1, 2, 3, 4, 1, 5, 6, 4, 1, 2, etc. The faces are as follows: two black triangles opposite each other; two black triangles in a different arrangement; two white triangles arranged as in face 1; two white triangles, arranged as in face 2; solid white; solid black.

The Same Way Fold can also be used, with the inner ﬂaps arranged differently: two opposite ﬂaps creased one way, the other two creased the other way. The result ﬂexes exactly the same as the one above, but with different designs: eight triangles of alternating color, circling about the center;

two black triangles opposite each other; solid black; solid black; same as the second face; solid black. This does not appear to offer as much variation due to the repetition involved. However, the ﬁrst face takes on a quite 126 R. E. NEALE different appearance when ﬂexed to the back of the model, as the eight triangles are rearranged in an interesting way — four triangles of alternating color circle about the center, with one triangle in each corner of the model.

Use of the Alternating Way Fold creates a puzzle version. Four faces are shown continually, and the other two are deceptively hard to ﬁnd, although no movement to a ring is required. The designs are these: checkerboard square; triangles in each corner so the center forms a white square; four large triangles alternating in color; checkerboard square; solid black; solid white. (This is my favorite of these four because the designs are handsome and the two solid color faces are a little tricky to ﬁnd.) The Alternating Way Fold can be used with the inner ﬂaps arranged differently, also. The designs on one pair of faces are different, two triangles on each face, but differing in both color and arrangement.

**Possibilities**

Other ﬂexagons can be constructed from the X-slit approach. For example, I constructed one to reveal the maximum of variety in design. This uses the fact that all ﬂexagons show some faces more than once. The second and third showings of the face reorient parts of it. So although my multiversion shows only six faces, these reveal seven different designs: solid color;

checkerboard of four squares; one triangle (out of eight); four faces each with different arrangements of three triangles.

Additionally, the methods given above can be combined to create still different design possibilities. For example, when a half of a corner is removed, instead of all or none of it, the design changes. For another example, combine a half-cross base with a slash-slit, and use the Same Way Fold.

And so on.

This manuscript was submitted for possible publication after the Gathering for Gardner in January of 1993, but written more than a year or two before that. One of the items was taught at the Gathering and appeared in the Martin Gardner Collection.

The Odyssey of the Figure Eight Puzzle Stewart Cofﬁn I became established in the puzzle business in 1971, hand-crafting interlocking puzzles in fancy woods and selling them at craft shows as fast as I could turn them out. What I lacked from the start was a simple interesting puzzle that could be produced and sold at low cost to curious children (or their parents) who could not afford the fancy prices of those AP-ART sculptures. To ﬁll this need, I turned my attention to topological puzzles, which almost by deﬁniton do not require close tolerances and are just about the easiest to fabricate. This led to a couple of not very original or interesting string-and-bead puzzles (the Sleeper-Stoppers). I also tried to come up with something in string and wire that could be licensed for manufacture. I may have been inspired in this by a neat little puzzle known as Loony Loop that was enjoying commercial success at the time. The three rather unexciting puzzles that came out of this (Lamplighter, Liberty Bell, and Bottleneck) are described in the 1985 edition of Puzzle Craft, so I will not waste space on them here, except to describe their common design feature.

**Take a foot or two of ﬂexible wire and form a loop at both ends, thus:**

Then bend the wire into some animated shape, with wire passing through the loops in various ways, such as in the very simple example below.

Finally, knot a sufﬁciently long loop of cord around some part of the wire and try to remove it (or try to put it back on after it has been removed).

Unlike the simple example shown above, some of these puzzles can be quite bafﬂing to solve.

Alas, none of these ideas enjoyed any success. (In a recent survey of my puzzle customers, they reported that topological puzzles in general were their least favorite of any puzzles I had produced.) The story does not end there, however. For a slight variation of this, during an idle moment one day I formed some wire into the ﬁgure-eight shape shown below, knotted a cord around it, and tried to remove it.

I soon became convinced that this was impossible, but being a novice in the ﬁeld of topology, I was at a loss for any sort of formal proof. I published this simply as a curiosity in a 1974 newsletter (later reprinted in Puzzle Craft). Some readers misread that purposely vague write-up and assumed that it must have a solution, which then left them utterly bafﬂed as to ﬁnding it. Royce Lowe of Juneau, Alaska, decided to add the Figure Eight to the line of puzzles that he made and sold in his spare time. When some of his customers started asking for the solution, he begged me for help.

Next it appeared in a 1976 issue of a British magazine on puzzles and games. The puzzle editor made the surprising observation that it was topologically equivalent to the Double-Treble-Clef Puzzle made by Pentangle and therefore solvable, since the Pentangle puzzle was. But a careful check showed that the two were not quite equivalent.

To add to the confusion, somewhat to my surprise, the puzzle appeared in Creative Puzzles of the World by van Delft and Botermans (1978) with a farce of a convoluted “solution” thrown in for added amusement. More recently, I received a seven-page document from someone in Japan, full of diagrams and such, purporting to prove that the puzzle was unsolvable.

The proof appeared to be a rather complicated, and I did not spend a lot of time trying to digest it. Over the intervening years, I had continued to

## THE ODYSSEY OF THE FIGURE EIGHT PUZZLE 105

received numerous requests for the solution, and by that time I was rather tired of the whole thing.When it comes to puzzles, it is often the simplest things that prove to have the greatest appeal, probably not even realized at the start. Whoever would have guessed that this little bent piece of scrap wire and loop of string would launch itself on an odyssey that would carry it around the world? I wonder if this will be the ﬁnal chapter in the life of the infamous Figure Eight Puzzle. Or will it mischievously rise again, perhaps disguised in another form, as topological puzzles so often do?

Metagrobolizers of Wire Rick Irby Many people love the challenge of solving a good puzzle. In fact, those who like puzzles generally like to solve just about any problem. Be it a paradox, a mathematical problem, magic, or a puzzle, the search for answers drives many of us on. Unlike magic or illusions with misdirection and hidden mechanisms, mechanical puzzles are an open book, with everything visible, all parts exposed ready for minute examination and scrutiny. In spite of this, the solutions can elude even the sharpest and quickest minds of every discipline.

Puzzles can go beyond an understanding of the problem and its solution, and here is where the separation between the common puzzler and (to borrow a phrase from Nob Yoshigahara) a “certiﬁable puzzle crazy” lies. The majority of mechanical puzzle solvers take the puzzle apart through a series of random moves with no thought given to the fact that this way they have only half-solved the problem. The random-move method will sufce for easy to medium puzzles but will do little or no good for solving the more difﬁcult ones. A “puzzle crazy,” on the other hand, will analyze the problem with logic and stratagem, then reason out the solution to include returning the parts to their original starting position. Regardless of one’s ability to solve them, puzzles entertain, mystify and educate, and the search for puzzling challenges will undoubtedly continue.

My own interest in puzzles began in early childhood, with the small packaged and manufactured wire puzzles available at the local 5 & 10-cents store. Although entertaining, they were never quite enough of a challenge to satisfy my hunger. Somewhere in the back of my mind I knew I could come up with better puzzles than were currently and commercially available. About twenty years after my introduction to those ﬁrst little wire teasers, a back injury from an auto accident and lots of encouragement from puzzle collectors brought the following and many other puzzle ideas to fruition.

Wire disentanglement puzzles are topological in nature and can vary widely in both difﬁculty and complexity of design. Wire lends itself very easily to topological problems because of its inherent nature to be readily 132 R. IRBY and easily formed into whatever permanent shapes may be necessary to present a concept.

I am frequently asked to explain the thought process involved in coming up with a new puzzle. Unfortunately, I really can’t answer. I neither know or understand the process of any creativity than to say that it just happens.

It is my suspicion that the subconscious mind is constantly at work attempting to ﬁt pieces of countless puzzles together; sometimes it succeeds! If you devote your mind to something, either you become good at it or you are devoting your mind to the wrong thing. A couple of examples of ideas that have “popped” out of my mind at various times are explained and illustrated below.

Many thanks must go to Martin Gardner as an inspiration to the millions enlightened by his myriad works. Thanks also to Tom Rodgers for his support of my work, for asking me to participate in “Puzzles: Beyond the Borders of the Mind,” and for presenting me with the opportunity to meet Martin Gardner.

**The Bermuda Triangle Puzzle**

Knowing the fascination that many people have with the somewhat mysterious and as-yet-unexplained disappearances of various airplanes and ships in the area known as the Bermuda Triangle and the Devil’s Triangle made naming this puzzle relatively easy. Often it is easier to come up with and develop a new puzzle idea than to give it a good and catchy name. This puzzle idea came to me as I was driving to San Francisco to sell my puzzles at Fisherman’s Wharf, in 1971 or 1972.

** The Bermuda Triangle PuzzleMETAGROBOLIZERS OF WIRE 133**

The object of the Bermuda Triangle is to save the UFO that is trapped in the puzzle, the UFO being a ring with an

**Abstract**

shape mated to it.

The puzzle is generally set up with the ring around the Bermuda Triangle, which is a triangular piece. The triangle can be moved over the entirety of the larger conﬁguration, taking the UFO with it as it moves. There are several places where the UFO may be separated from the triangle but only one place where the separation will allow the solution to be executed. Most of the large conﬁguration to which the triangle is attached is there simply to bewilder the would-be solver. The solutions to many puzzles can be elusive until the puzzle has been manipulated many times; although moderately difﬁcult for the average puzzler to solve initially, this one is relatively easy to remember once the solution has been seen. The Bermuda Triangle rates about a medium level of difﬁculty.