Tangram

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For other uses, see Tangram (album).
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A typical tangram construction
A typical tangram construction

Tangram (Chinese: 七巧板; pinyin: qī qiǎo bǎn; literally "seven boards of skill") is a dissection puzzle. It consists of seven pieces, called tans, which fit together to form a shape of some sort. The objective is to form a specific shape with seven pieces. The shape has to contain all the pieces, which may not overlap.

Contents

[edit] History

Wooden Tangram Set
Wooden Tangram Set

The Tangram very possibly originated from the yanjitu (燕几圖) furniture set during the Song Dynasty. According to historical Chinese records, the furniture set was originally a set of 6 rectangular tables.[citation needed] Later, an additional triangular table was added to the set, and people can arrange the 7 tables into a big square table. There is some variation to such furniture set during the Ming Dynasty, and later became a set of wooden blocks for playing.

According to other authors, the earliest reference to tangram appears in a Chinese book dated 1813, which was probably written during the reign of the Emperor Jiajing.[1]

The tans are often stored as a rectangle
The tans are often stored as a rectangle

A legend states that a servant of a Chinese emperor was carrying a ceramic tile, extremely expensive and extremely fragile. The servant tripped, shattering the tile. In a panic, the servant desperately tried to reassemble the tile into a square, but could not. He spent many days trying to reassemble the pieces into a square again, but could not, and instead created thousands of patterns and pictures during his attempts.

While the tangram is often said to be ancient, its existence in the Western world has been verified no earlier than 1800. Tangrams were brought to America by Chinese and American ships during the first part of the nineteenth century. The earliest example known is made of ivory in a silk box and was given to the son of an American ship owner in 1802.[citation needed]

The word "tangram" is built from TANG + GRAM. The word "Tangram" was first used by Thomas Hill, later President of Harvard, in his book Geometrical Puzzle for the Youth in 1848.

The author and mathematician Lewis Carroll reputedly was a great enthusiast of tangrams and possessed a Chinese book with tissue-thin leaves containing 323 tangram designs. Napoleon was said to have owned a Tangram set and Chinese problem and solution books while he was imprisoned on the island of St. Helena although this has been contested by Ronald C. Read. Photos are shown in "The Tangram Book" by Jerry Slocum.

In 1903, Sam Loyd wrote a spoof of tangram history, The Eighth Book Of Tan convincing many people that the game was invented 4,000 years ago by a god named Tan. The book included 700 patterns some of which are not possible. [2]

Traditional tangrams were made from stone, bone, clay or other easy to get materials. Nowadays they can be made from plastic, wood or other modern materials.

[edit] Mathematical proofs

Fu Tsiang Wang and Chuan-chin Hsiung proved in 1942 that there only existed 13 convex patterns (i.e. patterns with no recesses in the outline).[3]

Convex tangrams are very special and there are so few of them. Ronald C. Read in his book “Tangrams: 330 Puzzles” asked for any other special kinds of tangrams that would be more numerous than the convex ones, and yet not infinite in number. [4] He proposed to investigate the "snug tangrams" with the use of a computer.

An estimate of ten millions of configuration has been reported,[5] focusing on the fully matched patterns (all the pieces paired each other, with an edge and at least one vertex of the first piece matched to an edge and vertex of the second one), which are indeed a wider set than the Read's "snug tangram".

Further steps in counting the Tangram configurations have been attempted [6] in the One Billion Silhouettes Project.


[edit] The pieces

Tan construction
Tan construction

Sizes are relative to the big square, which is defined as being of width, height and area equal to \scriptstyle{1} .[7].

  • 5 right-angle triangles
    • 2 small (hypotenuse of \scriptstyle{1/2} and sides of \scriptstyle{1/2\sqrt{2}})
    • 1 medium size (hypotenuse of \scriptstyle{1/\sqrt{2}} and sides of \scriptstyle{1/2})
    • 2 large size (hypotenuse of \scriptstyle{1} and sides of \scriptstyle{1/\sqrt{2}})
  • 1 square (side of \scriptstyle{1/2\sqrt{2}})
  • 1 parallelogram (sides of \scriptstyle{1/2} and \scriptstyle{1/2\sqrt{2}})

Of these 7 pieces, the parallelogram is unique in that its mirror image cannot be obtained by rotation. Thus, it is the only piece that needs to be flipped when forming some silhouettes. Since there is only one such piece, every possible silhouette or its mirror image can be formed with a set of one-sided tangrams (for example, tangrams with a magnetic back that slide on a magnetic board).

[edit] See also

[edit] References

  1. ^ Chen, Zhongying (1999). Advances in computational mathematics: proceedings of the Guangzhou international symposium. New York, N.Y: Marcel Dekker, p.466. ISBN 0-8247-1946-8. 
  2. ^ Costello, Matthew J.. The Greatest Puzzles of All Time. New York: Dover Publications, p.45. ISBN 0-486-29225-8. 
  3. ^ Read, Ronald C.. Tangrams : 330 Puzzles. New York: Dover Publications, p.53. ISBN 0-486-21483-4. ,A Theorem on the Tangram, American Mathematical Monthly.
  4. ^ Read, Ronald C.. Tangrams : 330 Puzzles. New York: Dover Publications, p.55. ISBN 0-486-21483-4. 
  5. ^ TangMath, Ten Millions of Tangram Patterns, by F.Cocchini
  6. ^ Tanzzle, 1,000,000,000 silhouettes
  7. ^ "Tangram" by Enrique Zeleny, The Wolfram Demonstrations Project.

[edit] Further reading

  • Anno, Mitsumasa. Anno's Math Games (three volumes). New York: Philomel Books, 1987. ISBN 0399211519 (v. 1), ISBN 0698116720 (v. 2), ISBN 039922274X (v. 3).
  • Binary Arts. Tangram (puzzle game). 2003. ISBN 0-641-50914-6.
  • Botermans, Jack, et al. The World of Games: Their Origins and History, How to Play Them, and How to Make Them (translation of Wereld vol spelletjes). New York: Facts on File, 1989. ISBN 0816021848.
  • Crawford, Chris. Tangram Puzzles: 500 Tricky Shapes to Confound & Astound (Includes Deluxe Wood Tangrams). New York: Sterling Publishing Company, 2002. ISBN 0-8069-7589-X.
  • Dale Seymour Publications. ESS Tangram Cards: Grades K-8; The Elementary Science Study. Palo Alto, Calif., 1997. ISBN 0866514414.
  • Dale Seymour Publications. ESS Tangram Cards: Set II; The Elementary Science Study. Palo Alto, Calif., 1997. ISBN 0866514422.
  • Dale Seymour Publications. ESS Tangram Cards: Set III; The Elementary Science Study. Palo Alto, Calif., 1997. ISBN 0866514430.
  • Deutsch, E. S., and K. C. Hayes Jr. "A Heuristic Solution to the Tangram Puzzle", Machine Intelligence vol. 7, 1972, p. 205–240.
  • Dudeney, H. E. Amusements in Mathematics. New York: Dover Publications, 1958.
  • Elffers, Joost. Tangram: The Ancient Chinese Shapes Game. Penguin Books, 1976. ISBN 0-14-004181-8.
  • Elffers, Joost, and Michael Schuyt. Tangrams: 1,600 Ancient Chinese Puzzles. New York: Barnes & Noble Books, September 2001. ISBN 0-7607-2712-0.
  • ETA/Cuisenaire. The Super Source Tangrams Resource Book, Grades K-2. Vernon Hills, Illinois, 1996. ISBN 978-1-57452-015-6.
  • ETA/Cuisenaire. The Super Source Tangrams Resource Book, Grades 3-4. Vernon Hills, Illinois, 1996. ISBN 978-1-57452-016-3.
  • ETA/Cuisenaire. The Super Source Tangrams Resource Book, Grades 5-6. Vernon Hills, Illinois, 1996. ISBN 978-1-57452-017-0.
  • Ford, Barbara E. The Master Revealed: A Journey with Tangrams. Vallejo, Calif.: Tandora's Box Press, 1990. ISBN 0-9627337-8-4.
  • Ford, Barbara E. Tangrams: The Magnificent Seven Piece Puzzle. Vallejo, Calif.: Tandora's Box Press, 2003. ISBN 0-9627337-7-6.
  • Foster, Dr. T. E. Camella Runs Free: A Tangram Adventure. Charleston, South Carolina: Booksurge, 2007. ISBN 978-1-4196-6606-3.
  • Foster, Dr. T. E. Delightful Illusions (Or Not) With Tangrams. Charleston, South Carolina: Booksurge, 2007. ISBN 978-1-4196-7537-9.
  • Foster, Dr. T. E. The Legend of the Tangram Prince. Charleston, South Carolina: BookSurge, 2007. ISBN 1-4196-6165-5.
  • Foster, Dr. T. E. The Tangram ABC Book (black & white). ?: Printing Craft, 1975.
  • Foster, Dr. T. E. The Tangram ABC Book (color). Charleston, South Carolina: BookSurge, 2006. ISBN 1-4196-4915-9.
  • Foster, Dr. T. E. Then and Now on Old Macdonald's Farm: A Tangram Number Book. Charleston, South Carolina: BookSurge, 2006. ISBN 1-4196-4916-7
  • Foster, Thomas E. Tangram Patterns. Creative Publications, 1977. ISBN 0-88488-081-8.
  • Gardner, Martin. "Mathematical Games-on the Fanciful History and the Creative Challenges of the Puzzle Game of Tangrams", Scientific American Aug. 1974, p. 98–103.
  • Gardner, Martin. "More on Tangrams", Scientific American Sep. 1974, p. 187–191.
  • Gardner, Martin. The 2nd Scientific American Book of Mathematical Puzzles and Diversions. New York: Simon & Schuster, 1961. ISBN 0671245597.
  • Hartswick, F. Gregory. Adventure of the Beautiful Princess in Triangle Land. New York: Simon & Schuster Publishers, 1925.
  • Hirsch, Allan F. A Teacher's Guide to Tangram Mastery. Seattle: Alleyoop Enterprises, 1996.
  • Irvin, Barbara Bando. Geometry and Fractions with Tangrams. Vernon Hills, Illinois: Learning Resources, Inc., 1995(?). ISBN 1569119724.
  • Johnston, Susan. The Fun with Tangram Kit. Mineola, New York: Dover Publications. 1977. ISBN 0-486-23436-3.
  • Johnston, Susan. Tangram ABC Kit. Mineola, New York: Dover Publications, 1979. ISBN 0486238539.
  • Keen, Frank C. The tale of 2 bunnies (A tangram tale). Lulu.com, October 2007.
  • Keen, Frank C. The tale of the very, very weird cat (A tangram tale). Lulu.com, November 2007.
  • Kohner Bros. Inc. Pythagoras-A Game of 179 Puzzles.
  • Lehet, John L. A Sage's Journey: The Story of Tangrams. Waterford, Connecticut: MathMaverick Press, 1998.
  • Li, H. Y. Tangrams: I Ching Games of Duke Tan of Chou and C. C. T'ung. San Francisco: Cadleon Press, 1971.
  • Lindgren, Harry. "Tangrams", Journal of Recreational Mathematics 1969, p. 184–192.
  • Loyd, Sam. Sam Loyd's Book of Tangram Puzzles (The 8th Book of Tan Part I). Mineola, New York: Dover Publications, 1968.
  • Lyon, James. Tangram Box. Book-in-a-Box Publications, 2005.
  • Van Delft, Pieter, et al. Creative Puzzles of the World. New York: Harry N. Abrams, Inc., 1978. ISBN 0810907658, ISBN 0810921529.
  • Read, Ronald C. Tangrams: 330 Puzzles. New York: Dover Publications, 1965. ISBN 0-486-21483-4.
  • Seymour, Dale. Tangramath. Palo Alto, Calif.: Dale Seymour Publications Inc., 1971. ISBN 0884881482.
  • Slocum, Jerry, and Jack Botermans. The Book of Ingenious and Diabolic Puzzles. New York: Times Books, 1994. ISBN 0812921534.
  • Slocum, Jerry, et al. Puzzles of Old and New: How to Make and Solve Them. De Meern, Netherlands: Plenary Publications International (Europe); Amsterdam, Netherlands: ADM International; Seattle: Distributed by University of Washington Press, 1986. ISBN 0295963506.
  • Slocum, Jerry, et al. The Tangram Book: The Story of the Chinese Puzzle with Over 2000 Puzzles to Solve. New York: Sterling Publishing Company, 2003. ISBN 1-4027-0413-5.
  • Tompert, Ann. Granfather Tang's Story. New York: Crown Publishers, Inc., 1990. ISBN 051757487X, ISBN 0517572729.
  • Van Note, Peter. Tangrams: Picture Making Puzzle Game. Rutland, Vermont: C. E. Tuttle Co., 1966.
  • Wang, F. T., and C. C. Hsiung. "A Theorem on Tangram", American Mathematical Monthly vol. 49, 1942.