What is a Polyiamond?
Polyiamonds develop while putting equilateral triangles
together in such way they must have at least one side in common.
The Scottish mathematician T.H. O'Beirne suggested the
name polyiamonds in "New Scientist" 1961 [(1), page 164]. He called the
figures after the diamond.
||You can only form one figure with two or three triangles.
There are three figures of four triangles, the tetriamonds.
||There are four figures of five triangles.
Here is a triangle pattern for
||If you want to play with them, you should build a set.
Therefore you print a pattern of equilateral triangles,
mark the four figures in a size you like, glue them on cardboard and cut
Though the number of pieces is small,
you can design figures:
You can recognize:
1 intercity, 2 sphinx, 3 crooked tower, 4 trapezium without
a corner, 5 terraced houses, 6 motorboat, 7 motorboat with a peephole.
Even symmetric figures are possible:
The names of the figures go back to O'Beirne mentioned above.
It is worth working on the pieces with six triangles,
the hexiamonds. You have more pieces than with pentiamonds and therefore
more ways of playing. You can build them in the same way as described above.
There are 12 hexiamonds.
Playings on Hexiamonds
Pentominos are decribed
on another place of my homepage. Pentominos are pieces of five squares.
You can solve different problems like forming rectangles, new figures,
figures with holes, enlarged pentominos or rings. You can transfer these
problems to hexiamonds.
1st Problem: Parallelograms
||The Parallelograms 6*12 und 9*8 are possible and solved
on the left.
If you design your own figures, you
first must find out, whether you can solve them. You use the chessboard
method: You alternately colour all pieces and count the triangles
of each colour.
If you colour all 12 pieces, then 10 pieces have 3 white
and 3 black (grey) fields, 2 pieces have 4 or 2 black fields. There is
a statement for all pieces: They have the distribution 38+34 or 36+36.
The second sum comes, if you exchange the colours of the two pieces on
When you have designed a figure and have coloured it like
a chessboard, the distribution of the single pieces must be transfered
to the whole figure.
The following figure has the distribution 38+34. You
can solve it.
It isn't sure that a figure is always possible, if the
distribution is 38+34 or 36+36.
You only can say, maybe it is soluble.
Is there a solution? What is it?
3rd Problem: Rings
||You build a ring of all hexiamonds.
Then you should surround as many connected (white) triangles
Is the number 91 the maximum?
4th Problem: Enclosing
||You must use all pieces and surround as many single triangles
I found eight triangles in a first attempt.
5th Problem: Small
You needn't use all 12 pieces for building new figures.
||You form a star of eight hexiamonds or 48 triangles.
6th Problem: Duplicating
||You build a hexiamond with double magnification using
four pieces. Eight pieces are left.
Question: Can you build all hexiamonds like this?
7th Problem: Tripling
||You build a hexiamond with triple magnification using
nine pieces. Three pieces are left.
You can only do it with nine hexiamonds (Origin: Instructions
8th Problem: Figures
of the same hexiamonds
||You can build a larger hexiamond ("Sphinx") with four
The piece "yacht" can cover the whole plane. Question: Which
pieces also tile the plane?
||The small and large hexiamonds needn't be the same.
Example: Four pieces ("yacht") form a larger hexiamond
Manufacturer: Herbert Zimpfer, Metallwarenfabrik, 7586 Altschweier
||There was a famous puzzle with the name "Verhext" (Bewitched)
in Germany in the 1960s. It used all the 12 hexiamonds.
Professor Heinz Haber developped it and presented it
in detail on television and in his magazine "Bild der Wissenschaft".
The pieces had the names Kamm, Kirche, Pfeil, Feile,
Revolver, Haken, Hexagon, Segelboot, Schlange, Tanker, Pfeffermühle,
||Name and Manufacturer unknown
Sent by Bodo Schnell
Furthermore there are 66 octiamonds, 160 figures of 9 triangles,
448 figures of 10 triangles, and 1186 figures of 11 triangles.
There are 24 heptiamonds.
Craig Knecht sent me the following interesting email.
Hi Jurgen Koller,
I have ventured into the world
of polyiamonds. I have made some cartoons and tried to humanize the
In the cartoon below -
The fat bird cannot be tiled with
the sphinx tile.
The hexagon with and without the
fat bird can be tiled with the sphinx tile.
I just find that to be interesting
I found a proof that all hexagons
can be tiled with the theoretical maximum number of heptiamonds....
on the Internet top
Andrew Clarke (Die Poly-Seiten)
Andrew Clarke (Die Poly Pages)
Col. George Sicherman (Polyform Curiosities)
Ed Pegg Jr. (mathpuzzle.com)
Eric W. Weisstein (MathWorld)
N. J. A. Sloane (The On-Line Encyclopedia
of Integer Sequences)
Number of triangular
polyominoes, Number of one-sided
(1) Martin Gardner: Mathematisches Labyrinth, Braunschweig
1971 (ISBN 3-528-08402-2)
(2) Karl-Heinz Koch: ...lege Spiele, Köln 1987 (ISBN
(3) M.Odier, Y.Roussel: Trioker mathematisch gespielt,
Braunschweig, Wiesbaden 1979 (ISBN 3-5 28-08394-8)
(4) Zusammenlegspiele mit Quadraten und Dreiecken, Bild
der Wissenschaft 11/1965, Seite 946ff., Fortsetzung in Heft 12/1965
(5) Noch einmal: "Verhext", Bild der Wissenschaft 3/1967,
Feedback: Email address on my main page
page is also available in German.
2003 Jürgen Köller