What are Pentominos?
||You call the 12 figures, which you can make of five squares,
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.
The rectangles have 2339 solutions (6x10), 2 solutions
(3x20), 368 solutions (4x15), 1010 solutions (5x12).
The main problem of the pentomino 'research' is to combine
12 pieces to rectangles.
You can form four different rectangles:
You can form a rectangle 5x13, if you leave blank a pentimono
(5x13 = 65 = 60 + 5).
Building New Figures
The results are the figures:
You can design more figures beside rectangles. Best you
don't plan a pattern, but start working. Then it is easier to discover
There are no limits for your fantasy.
Figures with Holes
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).
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.
||You can imitate some compact pentominos in double size.
You need four pieces, eight ones are left.
Darian Jenkins sent me the
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 =
- Students of the Belgian school TID in Ronse and their
teacher Odette de Meulemeester have specialized in these problems.
You can make rings of pentominos, build bridges or make
other figures, in order to surround as many squares as possible.
- The ring on the left includes 120 squares. There
is a ring with 128 squares (URL: http://www.iread.it/lz/maximizing.html).
- The bridge on the right includes 251 squares.
There is a bridge with 278 squares (8).
Once Again: Rectangles
There are 84 squares inside on the left, 90 are maximum.
(Book 6, W.F.Lunnon)
||You can also lay pentominos, so that the inside or the
border form rectangles.
Both is possible, too.
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.
of Pentacubes top
The boxes have the measurements 3x4x5
(3940 solutions), 2x5x6 (264 solutions)
oder 2x3x10 (12 solutions).
The main problem is forming boxes.
You can lay three:
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).
You imitate a pentomino with double magnification and
triple height. A solution of the T-Pentomino follows.
There are no limits for your fantasy.
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.
Datas of the
Meaning of the numbers:
There is the formula e + a - k = 2. (Thanks to 6c,
7a, 7c, 7d from 99/00)
||V volume, O surface area, K sum
k number of edges, e number of corners,
number of sides
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
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
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.
- Lösung 6x10 - Applet Online
Dr. Nagy László
Das Element der Pentominos ist kein Quadrat mehr,
sondern ein Rhombus. Puzzle-Aufgaben mit jetzt 20 Pentominos
play pentacubes online
David J. Eck
Solver(8x8 with 4 holes), Applet
Eric W. Weisstein (MathWorld)
All 22 Pentominoes
Gerard's Home Page
Universal Polyomino Solver
Mathematics of Polyominoes
Reid's polyomino page
Miroslav Vicher (Miroslav Vicher's Puzzles Pages)
Snaffles home page
and Packing results
Leonid Mochalov [PUZZLES of LEONID MOCHALOV]
(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.
(5) R.Thiele, K.Haase: Der verzauberte Raum, Leipzig,
(6) Jens Carstensen: Legespiele, MU26:2 1980 (Seite 5
(7) Solomon W.Golomb: Polyominoes, Princeton, New Jersey
(8) Pieter Torbijn: Pentominos Bridges, Cubism For Fun
59, November 2002
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page is also available in German.
1999 Jürgen Köller