|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.
There are 32 half squares or 16 squares altogether.
|All seven tangram pieces consist of half squares with
||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
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
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
||[3sqrt(2)]x[3sqrt(2)]-squares are possible, if you leave
blank one triangle or two.
2nd problem: Filling given shapes
||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 mistake:
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 be alike.
An example by H.Dudeney
About 100 students (aged 11/12/13) got the task to design
birds by tangram pieces.
Here is a small selection of nice birds.
Not all the students became friends with tangram pieces:
hate tangrams ;-).]
(Thanks to 6b, 6c, 7a, 7c, 7d in 1999/2000)
1 the sides at the right angle
are horizontal or vertical
You can follow the positions of the sides of the half
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 2.
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.
'A figure is convex', means: If you choose any two points
inside the figure, the whole line between the points must also be inside
Proof by Fu Traing Wang and Chuan-Chih Hsiung in 1942 (Book
||Surprising: There are only 13 convex figures, you can
build from tangram pieces.
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 convex perimeter.
Bruno Curfs proved in (5) that 44-konvex is an upper
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 solutions.
Making of Tangram
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 the drawing.
Probably the tangram pieces are developed from cutting
a 4x4-square in pieces.
of the Tangram Game top
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 with them.
Tangram on the
Claus Michael Ringel
von Jos van Uden, Tangram-Spiel
von Serj Dolgav zum Herunterladen
Tangram for you
tangram mit einer
galerie von 75 exponaten
Andrew D. Orlov
Barbara E. Ford
Tangrams - The Magnificent
Seven Piece Puzzle
Cyberchase (Educational Broadcasting Corporation)
Degree Tutor – Guide to online colleges and distance learning
MILLIONS OF TANGRAM PATTERNS and more
Tangram for you
(1) Pieter van Delft, Jack Botermans: Denkspiele der
Welt, München 1998
(2) Karl-Heinz Koch: ...lege Spiele, Köln 1987 (dumont
(3) Rüdiger Thiele, Konrad Haase: Teufelsspiele,
(4) Joost Elffers, Michael Schuyt: Tangram, Dumont, Köln
(+ tangram pieces)
(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
Feedback: Email address on my main page
page is also available in German.
1999 Jürgen Köller