|What is Tangram?
Tangram is one of the most popular games to lay.
You put figures of 7 pieces together (five triangles, one square and
one parallelogram). You must use all pieces. They must touch but not overlap.
Main Problem top
There are 32 half squares or 16 squares altogether.
|All seven tangram pieces consist of half squares with this shape:
||You can build a 4x4-square with all the 16 squares. The main problem
of the 'tangram research' is building a square with all the 7 pieces.
You can also choose the smallest tangram piece, the blue triangle,
as the basic triangle. I took the half triangle as the basic form, because
the square built of all seven tangram pieces has the simple length 4.
The difference is: You have to change the rational und irrational length
of a side.
|Basic triangle on this page:
1st problem: New figures top
||You can invent new figures.
A figure is good, if you recognize it by seeing.
There are thousands of figures, which people have already built.
||[3sqrt(2)]x[3sqrt(2)]-squares are possible, if you leave blank one
triangle or two.
2nd problem: Filling given shapes top
||You must find the right position of the tangram pieces filling this
shape. (Solution in the end of this chapter)
3rd problem: How many possibilities are there to
lay the same figure? top
||You can lay the trapezium in two different ways.
||This trapezium is not a solution. If you lay this figure, you find
The yellow and the green piece are a little bit bigger.
You use the fact, that 4 and the triple of the square root of 2 (=4.24)
are about the same.
||Other paradoxes comparing two similar tangram figures, which seem to
An example by H.Dudeney
Tangram Birds top
About 100 students (aged 11/12/13) got the task to design birds by
Here is a small selection of nice birds.
Not all the students became friends with tangram pieces:
[= I hate tangrams
(Thanks to 6b, 6c, 7a, 7c, 7d in 1999/2000)
Classifying Figures top
1 the sides at the right angle are horizontal
You can follow the positions of the sides of the half squares.
2 the side opposite the right angle is
horizontal or vertical
3 mixture of 1 and 2
4 any position
If you have a mathematical point of view, you must allow only 1 and
Nearly all tangram figures belong to model 4. There are many nice and
expressive forms, because there are less rules. They are organized by topics.
Convex Figures top
'A figure is convex', means: If you choose any two points inside the
figure, the whole line between the points must also be inside the figure.
Proof by Fu Traing Wang and Chuan-Chih Hsiung in 1942 (Book 4)
||Surprising: There are only 13 convex figures, you can build from tangram
Grid Tangrams with
Convex Perimeter top
The figures are widened by little (white) triangles as necessary, so that
a convex figure develops. These triangles have the same size as the blue
tangram pieces. You count the triangles. Bird 1 needs 14 triangles and
is 14-convex. Bird 2 is 5-convex. The convex figures don't need a triangle.
They are 0-convex. You find all 133 (abstract) 1-convex tangram figures
and solutions in book 4.
||You find an interesting suggestion in book 3 and book 4 to classify
tangram figures. This only refers to 'mathematical' figures', which bird
1 and bird 2 (see above) stand for. You can lay them into a coordinate
system, so that the corners of the seven tangram pieces have integers as
coordinates. In other words: You can order the tangram pieces in a way
that the sides with the unit 1 are horizontal or vertical. The lines with
the unit (root of 2) are diagonal.
There is the problem to find figures with the largest
Bruno Curfs proved in (5) that 44-konvex is an upper limit.
Here are Bruno Curfs' seven 41-convex tangrams he found.
||I received more 41-convex tangrams:
8 from Ludwig Welther, 9 from Hartmut Blessing,
10 and 11 from Hannes Georg Kuchler.
||Daniel Gronau checked all possible grid tangrams by his computer. He
found out that 41-convex is the upper limit und that there are three more
Making of Tangram Pieces
You use it to make tangram pieces. You draw a 4x4-square with some diagonals
on plywood or on cardboard. Then you saw or cut the pieces as shown at
Probably the tangram pieces are developed from cutting a 4x4-square
Variants of the Tangram
You can make more tangram games, if you divide simple geometric figures
like square, rectangle or circle. The most famous are (1) "Pythagoras",
(2) "Kreuzbecher", (3) "Alle Neune", (4) "Circular Puzzle",
(5) "The Broken Heart", and (6) "The Magic Egg".
Here is a wide field for designing your own tangram pieces and playing
Tangram on the Internet
Claus Michael Ringel
Grimm's GmbH, Spiel & Holz Design
von Jos van Uden, Tangram-Spiel
von Serj Dolgav - zum Herunterladen
Tangram for you
tangram mit einer galerie von
Andrew D. Orlov
Barbara E. Ford
Tangrams - The Magnificent Seven
Cyberchase (Educational Broadcasting Corporation)
Degree Tutor – Guide to online colleges and distance learning
OF TANGRAM PATTERNS and more
Tangram for you
(1) Pieter van Delft, Jack Botermans: Denkspiele der Welt, München
(2) Karl-Heinz Koch: ...lege Spiele, Köln 1987 (dumont taschenbuch1480)
(3) Rüdiger Thiele, Konrad Haase: Teufelsspiele, Leipzig 1991
(4) Joost Elffers, Michael Schuyt: Tangram, Dumont, Köln 1997
(5) Bruno Curfs: Mathematical Tangram, CFF, newsletter of the "Nederlandse
Kubus Club" NKC, 65 (November 2004)
(6) Jerry Slocum, Dieter Gebhardt, Jack Botermans, Monica Ma, Xiaohe
Ma: The Tangram Book, 2003
[ISBN 1-4027-0413-5] Sterling Publishing
Feedback: Email address on my main page
page is also available in German.
1999 Jürgen Köller