Contents of this Page
What are Pentominos?
Building Rectangles
Building New Figures
Figures with Holes
Magnification Problems
Once Again Restangles
From the Pentomino to the Pentacube
Boxes of Pentacubes
Large Pentacubes
More Figures of Pentacubes
Datas of the Pentacubes
Making of Pentominos 
More Pentacubes
Pentominos on the Internet
To the Main Page      "Mathematische Basteleien"

What are Pentominos?
... You call the 12 figures, which you can make of five squares, pentominos. 
You must arrange the squares, so that they must have in common at least one side. 

The shapes are similar to capital letters, so they have letters as names.

Building Rectangles top
The main problem of the pentomino 'research' is to combine 12 pieces to rectangles. 
You can form four different rectangles:

The rectangles have  2339 solutions (6x10), 2 solutions (3x20), 368 solutions (4x15), 1010 solutions (5x12). 

You can form a rectangle 5x13, if you leave blank a pentimono (5x13 = 65 = 60 + 5).

Building New Figures top
You can design more figures beside rectangles. Best you don't plan a pattern, but start working. Then it is easier to discover new figures.

The results are the figures:

There are no limits for your fantasy. 

Figures with Holes top
You can form a chessboard 8x8, if you admit 4 holes. (8x8-4=60) (drawing 1). 
You can derive new problems: 
> Rectangles with isolated holes (drawing 2),
> Figures with as much as possible isolated holes (drawing 3). There are only two solutions with 13 holes  (book 2).


Magnification Problems top
Pentominos with triple magnification:
...... ...... You build a pentomino with triple magnification. You need nine pieces. Three pieces are left. 

...... It is more difficult not to use the pentomino concerned. 

You can skip one pentomino in a triple pentomino and fill up the rest with eight pentominos. Four Pentominos are left.

Double Pentominos:
You can imitate some compact pentominos in double size. You need four pieces, eight ones are left.

Darian Jenkins sent me the following datas.

Number of ways to duplicate a pentomino using four pieces 
Dup F =
Dup I =
Dup L =
Dup N =
Dup P =
Dup T =
Dup U =
Dup V =
Dup W =
Dup X =
Dup Y =
Dup Z =
Number of ways to triplicate a pentomino using any nine of the twelve pieces 
Trip F = 443 Trip I = 201 Trip L = 938 Trip N = 610 Trip P = 9144 Trip T = 382 Trip U = 444 Trip V = 482 Trip W = 202 Trip X = 20 Trip Y = 809 Trip Z =
Number of ways to triplicate a pentomino not using the piece being replicated 
Trip F = 125 Trip I = 19 Trip L = 113 Trip N = 68 Trip P = 497 Trip T = 106 Trip U = 48 Trip V = 63 Trip W = 91 Trip X = 15 Trip Y = 86 Trip Z =

Rings top
You can make rings of pentominos, build bridges or make other figures, in order to surround as many squares as possible.
Students of the Belgians school TID in Ronse and their teacher Odette de Meulemeester have specialized in these problems. You find better solutions on their web sites (URL below). 

Once Again: Rectangles top
...... You can also lay pentominos, so that the inside or the border form rectangles. 

Both is possible, too. 

There are 84 squares inside on the left, 90 are maximum. (Book 6, W.F.Lunnon)

From the Pentomino to the Pentacube   top
...... Pentomino pieces are mostly not two-dimensional, but are made of cubes and form pentacubes in the plain. They are easier to manage and make new 3D-puzzles possible.

Boxes of Pentacubes   top
The main problem is forming boxes
You can lay three:
The boxes have the measurements 3x4x5 (3940 solutions), 2x5x6 (264 solutions) oder 2x3x10 (12 solutions).

Large Pentacubes top
You imitate a pentomino with double magnification and triple height. A solution of the T-Pentomino follows.
Number of solutions: W (0), X(0), F(1),  T(3), Y(7), U(10), I(12), V(21), Z(24), N(51), L(99), P(1082).

More Figures of Pentacubestop
You can design more complicated 3-D-shapes and build them with pentominos. 
A tower with a hole in the middle follows as an example. 
There are no limits for your fantasy.

Datas of the Pentacubes top
Meaning of the numbers: 
V volume, O surface area, K sum of edges 
k number of edges, e number of corners, a number of sides
There is the formula e + a - k = 2.  (Thanks to 6c, 7a, 7c, 7d from 99/00) 

More Pentacubes top
... There are 17 three-dimensional pentacubes besides the standard ones. Five are symmetric with a plane (pink). The remaining pentacubes appear as pairs of  symmetric mirror solids. Three pairs have three cubes in line (blue), three only two cubes (green). 

Making of Pentominos top
If you want to play with pentominos you have to make them with your hands. 

Squares of cardboard will do because many problems are restricted to two-dimensional figures. 

You can make pentominos of cubes. You buy a length of wood which is square in cross-section, cut it into cubes and glue the cubes together. 

Another method is glueing dice. The best thing is to use a two component glue, because it needs time to harden. Then you can form the pentominos without having to hurry. 

A cheap method is making them from a sheet of paper. You must design a base of every pentomino and fold it. 

Pentominos im Internet top


Andrew Clarke

Pentominos - Lösung 6x10 - Applet Online

Dr. Nagy László
Pentomino HungarIQa
Das Element der Pentominos ist kein Quadrat mehr, sondern ein Rhombus. Puzzle-Aufgaben mit jetzt 20 Pentominos

Edith Stein Schule
Pentominos  (u.a. Parkette)

Thimo Rosenkranz

Pentomino, Polyomino


Andrew Clarke

Three Cube Puzzle

play pentacubes online

David J. Eck
Pentomino Solver(8x8 with 4 holes), Applet

Eithan Samara
Pentominoes-3D Solver

Eric W. Weisstein (MathWorld)
Pentomino, Polyomino

François Labelle
Unfolding All 22 Pentominoes

Gerard's Home Page
Gerard's Universal Polyomino Solver

Ken Zeltner
Pentomino Fuzion Puzzles

Kevin Gong
The Mathematics of Polyominoes

Michael Reid
Michael Reid's polyomino page

Miroslav Vicher (Miroslav Vicher's Puzzles Pages)

Snaffles home page
Pentomino Relationships

Torsten Sillke
Tiling and Packing results

Pentomino, Polyomino


Puzzles with Polyominoes

References  (German) top
(1) Martin Gardner: Mathematical Puzzles & Diversions, New York 1959
(2) bild der wissenschaft 7/1976
(3) Pieter van Delft, Jack Botermans: Denkspiele der Welt, München 1980
(4) Martin Gardner: Bacons Geheimnis, Frankfurt a.M. 1986 (Polywürfel)
(5) R.Thiele, K.Haase: Der verzauberte Raum, Leipzig, 1991 
(6) Jens Carstensen: Legespiele, MU26:2 1980 (Seite 5 bis 36)
(7) Solomon W.Golomb: Polyominoes, Princeton, New Jersey 1994 (ISBN0-691-08573-0)

 Feedback: Email address on my main page

This page is also available in German.

URL of my Homepage:

©  1999 Jürgen Köller