Geometric Puzzles in the Classroom

superTangram 14-piece hexagon

by Henri Picciotto
This page includes some background information about my puzzle books, some articles, puzzles, and activities you can print for yourself or duplicate for your students, and a few links to relevant Web sites.

(Parts of this Web page are adapted, with permission, from articles I wrote in 1989 for Michael Keller's games and puzzles 'zine.)

Article outline:
Polyforms
Pentominoes
SuperTangrams
Polyarcs

I have found geometric puzzles to be an excellent springboard for mathematics lessons, as they are interesting to both students and teachers. They lend themselves to teaching some specific concepts as well as to building students' spatial sense and problem-solving skills. Moreover they show to a wide range of learners that exploring mathematics can be rewarding, irrespective of any "practical" application. As a result, geometric puzzles have consistently found their way into my classes, my workshops, and my books. In fact they were how I first broke into print.

There are many geometric puzzles that can be used in the classroom, such as for example tangrams and rep-tiles as presented in my book Geometry Labs, or Rubik's cube. In this article, I will concentrate on polyforms.

If you use a SmartBoard, you can download this gallery of convenient tangram, pentomino, and supertangram pages.

One-inch graph paper is useful for many of the activities described here.

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Polyforms

Polyforms are the shapes you get by combining multiple copies of a given "form" (usually copies of a given polygon, attached to each other edge-to-edge.) The best known example are the polyominoes, which are created by combining squares, and the best known among polyominoes are the shapes obtained by combining four squares. Those shapes are known as tetrominoes, and are the basis of the electronic game of Tetris.

the five tetrominoes

If you think of them as puzzle pieces you can pick up and turn over, there are five tetrominoes, as shown in the figure. (The shapes can be remembered with the mnemonic "square lint".) But if the pieces are one-sided and have to remain in the plane (as in Tetris), there are two more:

the seven tetris pieces

More on polyforms:

Polyominoes, by Solomon W. Golomb (the man who coined the word,) appeared in 1964. It is out of print, but the revised and expanded second edition: Polyominoes: Puzzles, Patterns, Problems, and Packings, is available from Princeton University Press, NJ, 1994. This is the polyform bible.

Several of Martin Gardner's books of Scientific American articles include articles on polyforms and related topics in recreational mathematics. (Look for polyaboloes, polycubes, polyhexes, polyiamonds, and polyominoes in the Gardner Index.)

The Journal of Recreational Mathematics (at university libraries) also includes articles on this topic, as does Michael Keller's games and puzzles 'zine.

On the Web, many sites devote substantial space and links to polyforms -- many more than I have time to keep up with. A search will turn them up. A few places to start are:

In the classroom:

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Pentominoes

Pentominoes are made of five squares, and are the subject of many, many classic puzzles.

the pentominoes

(The mnemonic "FLIP'N TUVWXYZ" is helpful in finding out quickly which piece is missing in an incomplete set.)

The earliest pentomino puzzle I know of is in H. E. Dudeney's 1919 book, The Canterbury Puzzles (available from Dover Publications), pages 119-121. In Arthur C. Clarke's science-fiction book Imperial Earth (Ballantine Books, 1976), Chapter 7 is mostly about a ten-year-old's exciting encounter with pentomino puzzles. "What had at first seemed merely a childish game had opened endless vistas and horizons... there was something strange here, something magical. It was Duncan's first glimpse of the power of intuition..."

More on pentominoes:
What rectangles can be simultaneously covered with pentominoes from one set? Find the answer in Simultaneous Pentomino Rectangles.

In the classroom:

(Where to get these materials.)
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SuperTangrams

SuperTangrams are the polyforms obtained by combining four unit isosceles right triangles snugly in every possible way. There are 14 superTangrams. Martin Gardner wrote about them under the name "tetraboloes" (in The Mathematical Magic Show, chapter 11). They are also known as "tetratans", a word I coined. The "superTangram" name obscures the fact that the figures are polyforms, but it does reveal the geometric connection with the classic tangram puzzle: both tangrams and superTangrams are made from right isosceles triangles, and in fact all three medium-sized tangram shapes are also superTangrams.

SuperTangram puzzles are more difficult than pentomino puzzles, and much more difficult than tangram puzzles. In SuperTangrams for Beginners 2, I proposed a notation for the 14 pieces. (Mnemonic: "BaCH, PRouST", and/or "BeaCH PReSTo".)

the supertangrams

A 14-piece rectangle is not possible (see Gardner). The proof generalizes to any rotationally symmetric shape (not, as Gardner claimed incorrectly, to bilaterally symmetric shapes). In 1989, I found and published two bilaterally symmetric convex 14-piece shapes:

convex, symmetric, 14 pieces

I also found another symmetric 14-piece shape, as did Ed Pegg (see this page on Michael Keller's site.) Finally, Miroslav Vicher found six other 14-piece convex figures through a computer search.

In the classroom:

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Polyarcs

Consider the two pieces that are created by drawing a quarter circle with radius one, centered at the corner of a unit square. Call these pieces monarcs. Their average area is 1/2.

monarcs

Two monarcs can be combined to create diarcs in seven distinct ways. The average area of the diarcs is 1.

diarcs

The total area of all of the monarcs and diarcs is 8. They can be combined into a 2x4 rectangle, or a pleasingly symmetric curvilinear figure:

monarc-diarc figures

There are 22 triarcs, with average area 1-1/2, and total area 33. In combination with the diarcs the area would be 40. All the usual polyform questions can be asked: What symmetric shapes can be constructed using all or some of the triarcs? All or some of the diarcs and triarcs?

In contrast to polyominoes, which have been researched extensively, little is known about polyarcs. If you or your students discover interesting polyarc figures, please send me copies of your solutions! Several puzzle enthusiasts did just that, as you can see below.

More on polyarcs:

You can purchase a beautiful translucent plastic set of diarcs and triarcs from Kadon Enterprises.

Diarcs and Triarcs, by Michael Keller.

Hundreds of solutions to diarc puzzles, by Bernd Karl Rennhak.

Tetrarcs and Pentarcs, by Brendan Owen.

Hundreds of monarc/diarc figures, by Alex Link.

A zipped Word file of the monarcs and diarcs, by Alex Link. Using Word's drawing tools, they can be dragged, flipped and rotated into pleasing shapes.

In the classroom:

A Polyarcs Lab for your students. Prerequisites: Students need to know how to find the area and perimeter of a circle.

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Related pages on this site:

Polyomino Lessons
Manipulatives
Geometry Labs
Geometry
Geometric Transformations

Elsewhere:
Working with Pentominoes by Henri Picciotto
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