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

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in Japan. She was the author of a book entitled Makura no Soshi (Pillow Book:

A Collection of Essays).

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shapes are inaccurate. The puzzle was introduced to the world by Martin Gardner in his book Time Travel ½, and he elaborated as follows.

“Shigeo Takagi, a Tokyo magician, was kind enough to send me a photocopy of this rare book. Unlike the Chinese tans, the Shonagon pieces will form a square in two different ways. Can you find the second pattern?

The pieces also will make a square with a central square hole in the same orientation. With the Chinese tans it is not possible to put a square hole anywhere inside a large square.”

A square with a center hole

I know of two other books that are collections of patterns of the Sei Shonagon pieces. In about 1780, Takahiro Nakada wrote a manuscript entitled “Narabemono 110 (110 Patterns of an Arrangement Pattern),” and Edo Chie-kata (Ingenious Patterns in Edo) was published in 1837. In addition, I possess a sheet with wood-block printing on which we can see patterns of the Sei Shonagon pieces, but its author and date of publication are unknown.

–  –  –

Do you want to create, or better generate, a new two-dimensional puzzle?

Nothing could be easier! Just take one of the many well-known puzzles of this type and submit it to a geometrical transformation. The result will be a new puzzle, and — if you do it the right way — a nice one. As an example, let us take the good old Tangram and try a very simple transformation, a linear extension (Figure 1).

Figure 1. Tangram submitted to a linear extension.

The result is a set of pieces for a new puzzle, a Tangram derivative called Trigo-Tangram.

Ô While the tangram pieces can be arranged in an orthogonal grid, the pieces of our new puzzle, due to ¿ as extension factor, fit into a grid of equilateral triangles, a trigonal grid. That’s why I gave it the name TrigoTangram.

Puzzles generated by geometrical transformations I call derivatives ; our new puzzle is a Tangram derivative, one of an infinite number. Depending on the complexity of the transformation, certain properties of the original puzzle are preserved in the derivative, others not. Tangram properties preserved in Trigo-Tangram are the linearity of the sides, the convexity of the pieces, and the side length and area ratios. Not preserved, for example, 100 B. WIEZORKE Figure 2. Suitable problems for a puzzle derivative can be found by transforming problems of the original puzzle. For Trigo-Tangram, two problems can be generated by extending this original Tangram problem in two different directions. The transformation of the solution can be a solution for the derivative (right image) or not (upper image).

are the angles and the congruence of both the two big and the two small triangles.

For our derivatives, we do not only need the pieces, we need problems as well. Problems? No problem at all! We just take some figures for the original puzzle and submit them to the same transformation as the puzzle itself. The images of the figures will be suitable problems for the derivative.

In Figure 2 this is done for a Tangram problem and its solution. As you can see, the corresponding linear extension can be performed in two different directions, which gives us two different problems. And you can see another important fact: The transformation of the solution does not always render a solution for the derivative.

Figure 3 shows some problems for Trigo-Tangram. So, take a piece of cardboard, copy and cut out the pieces, and enjoy this new puzzle.

Here is another way to have fun with a puzzle and its derivative: Take the original Tangram, lay out a figure, and try to make a corresponding figure with the pieces of Trigo-Tangram. In many cases, you get funny results.

Figure 4 shows how a Tangram bird stretches its neck and how a Tangram cat turns into the Pink Panther.


Figure 3. Some problems for the Trigo-Tangram puzzle.

Transforming two-dimensional puzzles provides endless fascination. Just look what happens if you rotate the Tangram in Figure 1 by ¼ Æ and then perform the transformation. The square and parallelogram of the original are transformed into congruent rhombi, which makes the derivative less convenient for a puzzle. You obtain better results by applying linear extensions diagonally (Figure 5). Since in this case the congruence between the two large and two small Tangram triangles is preserved, the two puzzles

–  –  –

Figure 5. Transforming the Tangram by diagonal extensions leads to two derivatives which are more Tangram-like than the one shown in Figure 1.

generated are even more Tangram-like than the Trigo-Tangram. To create problems for these derivatives is left to the reader. One set of problems will serve for both of them.

I hope I have stimulated your imagination about what can be done by transforming puzzles so you may start your own work. Take your favorite puzzle, and have fun generating individual derivatives for yourself and your friends!

Polly’s Flagstones Stewart Coffin I wish to report on some recent correspondence with my good friends the Gahns in Calcutta. You may remember meeting Paul in The Puzzling World of Polyhedral Dissections. His wife Polly is an avid gardener. She presented me with the following problem.

Polly places precisely fitted flagstones around various plantings in her garden in order to suppress weeds. She and Paul have a bent for geometrical recreations, so they are always on the lookout for creative and original solutions to their various landscaping projects. She had one large stone that was perfectly square. She asked me if it were possible to cut the stone into four pieces that could be arranged to form a somewhat larger square perimeter enclosing a square hole having sides one quarter those of the original stone. The simple solution to this problem which I then sent to her, a classic dissection of the square, is shown below.

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Now she posed another problem. She decided that she preferred mostly rectangular rather than square planting spaces. Suppose we were to consider starting with a large rectangular stone, which she proposed that we cut into four pieces, but this time to form a rectangular perimeter enclosing a rectangular planting space. And for good measure, to allow for more flexibility in landscaping, how about a scheme whereby the stones could be rearranged to form any one of three different-sized rectangular openings, each one enclosed by a rectangular perimeter. This required a bit more reection than the first problem, but after tinkering for a while with paper, pencil, and scissors, I came up with the scheme shown below.

Almost any rectangle dissected symmetrically by two mutually perpendicular lines that touch all four sides will produce four quadrilaterals that can be rearranged to create the required three different rectangular holes enclosed by rectangles. One of these rectangular holes will have the same shape as the original rectangle and will be surrounded by a rectangle also having that same shape. With the dimensions shown, the medium-sized rectangular hole will have dimensions exactly one quarter those of the original stone. The smaller and larger holes have dimensions that are irrational.

If you think that Polly was satisfied with this solution, then you don’t know Polly very well. She immediately wrote back and suggested that it was a shame to cut up two beautiful large stones when one might do, cut into four pieces, which could then be rearranged to create a square hole within a square enclosure or any one of three different rectangular holes within rectangular enclosures. It was then I realized that from the start she POLLY’S FLAGSTONES 105 had just been setting me up. I decided to play into it, so I wrote back and asked what made her so sure it was even possible. Sure enough, by return mail arrived her solution, shown below.

For the purpose of this example, let the original square stone ABCD be four feet on a side. Locate the midpoint E of side BC. The location of point F is arbitrary, but for this example let it be one foot from A. Draw EF. Subtract length CE from length DF, and use that distance from F to locate point G.

Bisect EG to locate H, and draw JHK perpendicular to EHF (most easily done by swinging arcs from E and G ).

Don’t ask my why this works, but it does. The actual arrangements of the pieces for the four different solutions, one square and three rectangular, are left for the reader to discover. One of the rectangles has a pair of solutions — the others are unique. With these dimensions, the square hole will be one foot square. For added recreation, note that the pieces can also be arranged to form a solid parallelogram (two solutions), a solid trapezoid (two solutions), and a different trapezoid (one solution).

An interesting variation is to let points A and F coincide, making one of the pieces triangular. The same solutions are possible, except that one of the trapezoids becomes a triangle.

Now what sort of scheme do you suppose Polly will come up with for that other stone, the rectangular one?

Those Peripatetic Pentominoes Kate Jones This recital is yet another case history of how the work of one man—Martin Gardner— has changed the course and pattern of a life.

In 1956 I received a gift subscription to Scientific American as an award for excellence in high school sciences. My favorite part of the magazine was the “Mathematical Games” column by Martin Gardner. The subscription expired after one year, and I was not able to renew it. It expired one month before the May 1957 issue, so I did not get to see the historic column introducing pentominoes (Figure 1) to a world audience.

Figure 1.

That column was inspired by a 1954 article in the American Mathematical Monthly, based on Solomon Golomb’s presentation in 1953 to the Harvard Mathematics Club. Golomb’s naming of the “polyomino” family of shapes and their popularization through Martin Gardner’s beloved column, focusing especially on the pentominoes, created an ever-widening ripple effect.

Arthur C. Clarke, in his mostly autobiographical Ascent to Orbit, declares himself a “pentomino addict”, crediting Martin Gardner’s column as the source. Thwarted from including pentominoes in the movie 2001: A Space Odyssey, as the game HAL and Bowman play (the film shows them playing chess), Clarke wrote pentominoes into his next science fiction book, Imperial Earth. Later editions of the book actually show a pentomino rectangle on the flyleaf.

In late 1976, a group of expatriates stationed in Iran took a weekend trip to Dubai. As we loitered around the airport newsstand, a paperback 108 K. JONES rack with a copy of Imperial Earth caught my eye. A longtime Clarke fan, I bought the book and soon caught pentomino fever. It was a thrill to find mention of Martin Gardner in the back of the book.

Playing first with paper and then with cardboard was not enough. I commissioned a local craftsman to make a set of pentominoes in inlaid ivory, a specialty of the city of Shiraz. This magnificent set invited play, and soon friends got involved. Exploration of the pentominoes’ vast repertoire of tricks was a fine way to spend expatriate time, and inevitably a dominotype game idea presented itself to me, to be shared with friends.

Fast forward to December 1978, when the Iranian revolution precipitated the rapid evacuation of Americans. Back home I was in limbo, having sold my graphics business and with no career plans for the future. My husband’s job took care of our needs, but idleness was not my style. A friend’s suggestion that we “make and sell” that game I had invented popped up just then, and after some preliminary doubts we went for it.

By fall of 1979 a wooden prototype was in hand, and the pieces turned out to be thick enough to be “solid” pentominoes. Well, of course that was the way to go! A little research turned up the fact that “pentominoes” was

Figure 2. Super Quintillions.THOSE PERIPATETIC PENTOMINOES 109

a registered trademark of Solomon Golomb, so we’d have to think of another name. Fives quints quintillions! It was with great pride, joy and reverence that we sent one of the first sets to the inspiration of our enterprise, Martin Gardner. And it was encouraging to us neophyte entrepreneurs when Games magazine reviewed Quintillions and included it on the “Games 100” list in 1980.

We had much to learn about marketing. The most important lesson was that one needed a product “line”, not just a single product. And so Quintillions begat a large number of kindred puzzle sets, and most of them sneak in some form of pentomino or polyomino entity among their other activities. Here (chronologically) are the many guises and offspring of the dozen shapes that entered the culture through the doorway Martin Gardner opened in 1957.½ Super Quintillions: 17 non-planar pentacubes (plus one duplicate piece to help fill the box). These alone or combined with the 12 Quintillions blocks can form double and triplicate models of some or all of the 29 pentacube shapes (see Figure 2).

Leap: a ¢ grid whereupon polyomino shapes are plotted with black and white checkers pieces in a double-size, checkerboarded format. The puzzle challenge: Change any one into another in the minimum number of chess knight’s moves, keeping the checkerboard pattern (Figure 3).

–  –  –

Void: a ¢ grid on which a “switching of the knights” puzzle is applied to pairs of polyomino shapes from domino to heptomino in size, formed with checkers (Figure 4). What is the minimum number of moves to exchange the congruent groups of black and white knights?

–  –  –

Quintachex: the pentominoes plus a ¾ ¢ ¾ square checkerboarded on both sides (different on the two sides). The pieces can form duplicate and triplicate models of themselves, with checkerboarded arrangements (Figure 5).

–  –  –

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