# How Tangrams Work

Whether it's rock 'n' roll in the '50s, or violent video games in the 2000s, people have always found some new fad to use as a scapegoat for the supposed laziness and corruption of their generation. It might seem quaint to think that tangrams, simple Chinese puzzles involving rearranging small geometric tiles, could ever have filled that role in society. But tangrams were such a huge craze in Europe in the early 19th century that a French newspaper cartoonist once lampooned the fad with a drawing of a young "bourgeois" couple ignoring their screaming baby while they worked out a tangram problem [source: Slocum]. Today, tangrams can be an educational tool for the classroom, or just a fun brain teaser for the geometrically inclined. While tangrams' days of corrupting the youth may be over, they have endured as a favorite puzzle for people of all ages.

Tangrams are part of a type of puzzle that mathematicians and puzzle experts call dissection puzzles, puzzles made of cut out shapes that can be combined to form other shapes or designs. The most famous type of dissection puzzle is a jigsaw puzzle. But tangrams are different -- instead of one design cut up into many pieces with one solution, tangrams have just a few pieces that can be rearranged into a huge number of designs. A tangram set is composed of seven shapes cut out of a square. These tiles, or tans, include:

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• two large triangles
• two small triangles
• one medium triangle
• one parallelogram (or rhomboid)
• one square

Tangram puzzles come in the form of silhouettes or outlines that are constructed by combining all seven tans. The object of the game is to re-create these shapes by figuring out the orientation of each tan. In books of puzzles, the solutions (or completed tangrams) show the outlines of the individual tans, but the puzzles don't. The challenge is for you to re-create the patterns by arranging the tans, following three simple rules: Each completed tangram must contain all seven tans; the tans cannot overlap each other; and the tans have to form a continuous shape, with every tan touching at least one other tan (touching only by the very tip of one corner is allowed).

These rules allow for a puzzle game that is deceptively simple and infinitely complex. Continue to the next page to find out when tangrams were invented.

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Contents

## Tangram History

Tangrams' true origins are obscure. Historians don't really know exactly when they were invented. The earliest record of the puzzle dates to around 1796, to a book that is mentioned in the historical record, but has never been found [source: Danesi]. Extant tangram sets have been dated as far back as 1802, and a Chinese book of tangram problems from 1813 has also been discovered [source: Slocum].

Whenever the modern form of the tangram was invented, the puzzle has its roots in Chinese mathematical tradition going back centuries. As long ago as the 3rd century B.C., Chinese mathematicians would study geometric principles by manipulating cut outs of various shapes. In fact, the Chinese used this method to deduce what Europeans call the Pythagorean theorem, the relationship between the sides and the hypotenuse of a right triangle. Historians have surmised that tangrams were likely developed from this type of problem solving [source: Slocum].

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Whatever the truth about tangrams, the myths and legends about their history are far more interesting. Most of these -- for example, the story that a mythical god named Tan invented the shapes, and used them to communicate a creation story in a set of parchments written in gold -- can be traced back to a writer and puzzle inventor named Sam Loyd. Loyd's 1903 book, "The 8th Book of Tan," weaved this and other tall tales about the history of tangrams. Loyd made those stories up, and probably expected his readers to be in on the joke [source: Slocum and Hotermans]. But to this day, some of Loyd's "history" shows up in otherwise factual sources.

Loyd's book rode a global wave of popularity for the tangram at the time. Almost as soon as tangrams spread from China to Europe and the United States, around 1818, they became a sensation. Books of puzzles and tile sets made of polished wood or ornately carved ivory became extremely popular in Germany, France, England, Italy and the United States.

Just like the origin of the puzzle itself, the origin of the name "tangram" is hard to pin down. At first, it was simply called "The Chinese Puzzle." The name tangram came later. Some of the theories include that it derived from the English word "trangam" (which means "trinket"). According to others, the word is a portmanteau of "Tang," a historical Chinese dynasty, and "gram," which means a figure or drawing [source: Grunfeld].

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## Tangrams and Mathematics

Tangrams have remained popular for so many years partly because they are so simple and at the same time so complex. In other words, because the individual tans are extremely simple shapes, an almost infinite number of combinations can be derived from them. In fact, there are more than 1 billion possible combinations that can be made with the seven tans [source: Cocchini].

The tans themselves are based on some very basic geometric principles. Each tan can be divided into several component triangles, each one a right isosceles triangle with a hypotenuse equal to √2 units, and two sides that measure 1 unit. (That unit can be inches, centimeters, feet, meters or even a made-up unit, because the shapes are based on proportional, not numerical, measurements).

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For example, the small triangles in the set are composed of two base triangles lined up side-by-side. The square is made up of two base triangles joined at the hypotenuse, and so on. To draw a set of tangrams, you can simply draw a square, superimpose a 4x4 grid over it, divide each square into two triangles, and then trace out the shapes along the borders of those triangles so that they match a tangram template. It doesn't matter what units you use to draw the grid, as long as it is perfectly square.

Often, tangram puzzles take on a shape, like a cat, a person or a sailboat. When it comes to these freeform shapes, there are potentially infinite combinations (especially when you factor in nonsense shapes that don't necessarily look like anything). However, there are some mathematical categories of figures that have set rules. These are easier to define and count.

Mathematical figures are those whose base triangles can all be lined up to a square grid. In other words, each shape is aligned so that at least one of its sides is perfectly horizontal or vertical [source: Koller]. With fully matched figures, every tan has at least one of its edges and one of its corners, or vertices, matched to at least one other tan. That is, there aren't any dangling pieces whose outlines can be easily identified. There are also fully aligned figures that can have dangling pieces, but at least one of the edges of any dangling tan has to form a continuous line with the border of the figure [source: Cocchini].

One specific subset of fully matched figures that mathematicians have studied is convex figures. These silhouettes are convex polygons -- shapes with interior angles all less than 180 percent. An easy way to tell if a polygon is convex is to draw a line between any two angles of the shape. If all of those lines either completely fit inside the figure, or perfectly match one of its borders, the shape is convex. Believe it or not, there are only 13 convex polygons that can be made out of the seven tans [source: Wang]. By contrast, the tans can form more than 10 million fully matched shapes [source: Cocchini].

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## Tangram Shapes and Patterns

Without the constraints that define some of the mathematical tangram patterns, there are a seemingly infinite number of possibilities. Beginning with the first books from China, tangram problems took on these more whimsical shapes, imitating a huge range of objects. Tangram patterns take the form of animals, buildings, household tools, people and vehicles. Even if it takes a little imagination to see the cat peering back at you from that blocky, triangular outline, that's part of the fun.

The only proven strategy for solving tangram problems is trial and error -- rearranging the shapes in multiple combinations until the answer strikes you. But, there are a few tips to solving the puzzles that you'll find in books, and these days, in collections of problems online.

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First, it's always easier to begin by identifying any dangling pieces -- the tans whose outlines are either completely exposed, or exposed enough that no other tan could take its place [source: Koller]. Of course, some of the tans are interchangeable. The two triangles can form the same shape as the parallelogram or the square, for example, so that dangling cat's tail might not be as easy to fill in as you think. It's also helpful to make note of any corners that stick out of the figure. An exposed triangular edge would eliminate the square from fitting in that spot, for instance.

The most difficult puzzles to solve are those that have regular edges with no corners or edges exposed [source: Koller]. For example, the convex polygons discussed on the previous page are notoriously tricky to solve. Probably the most difficult problem is forming the perfect square [source: Koller]. Since most sets of tangrams are sold pre-assembled as squares, most tangram players will have to face that particular challenge every time they put their tiles back in the box. Representational figures (animals, buildings, etc.) tend to be easier, since they have more jutting pieces that form ears, legs and chimneys. Read on for more information on tangrams, and to find Web sites where you can make and solve your own problems.

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## Lots More Information

### Sources

• Cocchini, Franco. "Ten Millions of Tangram Patterns." Tanzzle.com. (July 20, 2011) http://www.tanzzle.com/tangmath/BillionPatterns.htm
• Coffin, Stewart T. "The Puzzling World of Polyhedral Dissections." johnrausch.com. 1998. (July 20, 2011) http://www.johnrausch.com/PuzzlingWorld/contents.htm
• Danesi, Marcel. "The Puzzle Instinct." Indiana University Press. 2002.
• Grunfeld, Frederic V. ed. "Games of the World." Holt, Rinehart and Winston. 1975.
• Koller, Jurgen. "Tangram." Mathematische Basteleien. 1999. (July 20, 2011) http:/www.mathematische-basteleien.de/tangrams.htm
• Read, Ronald C. "Tangrams -- 330 Puzzles." Dover Publications, Inc. 1965.
• Sarcone, Gianni A. "Tangram, the Incredibly Timeless Chinese Puzzle." Archimedes-lab.org. (July 20, 2011) http://www.archimedes-lab.org/tangramagicus/pagetang1.html
• Slocum, Jerry. "The Tangram Book." Sterling Publishing Co., Inc. 2001.
• Slocum, Jerry and Jack Hotermans. "Puzzles Old and New: How to Make and Solve Them." University of Washington Press. 1986.
• Wang, Fu Traing and Chuan-Chih Hsiung. "A Theorem on the Tangram." The American Mathematical Monthly. Vol. 49, no. 9. Page 596-599. November 1942. http://www.jstor.org/stable/2302240

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