...	You call the 12 figures, which you can make of five squares, pentominoes.  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:

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.

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).

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Magnification Problems top
Pentominoes with triple magnification:

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You build a pentomino with triple magnification. You need nine pieces. Three pieces are left.

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It is more difficult not to use the pentomino concerned.

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

.......

You can imitate some compact pentominoes in double size. You need four pieces, eight ones are left.

Number of ways to duplicate a pentomino using four pieces 

Dup F = 1

Dup I = 2

Dup L = 8

Dup N = 7

Dup P = 50

Dup T = 1

Dup U = 7

Dup V = 0

Dup W = 5

Dup X = 0

Dup Y = 2

Dup Z = 7

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 = 395

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 = 131

Rings top
You can make rings of pentominoes, build bridges or make other figures, in order to surround as many squares as possible.  - Students of the Belgian school TID in Ronse and their teacher Odette de Meulemeester have specialized in these problems. 
   (URL http://pentomino.classy.be/indexnl.html). 
 - 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 top

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You can also lay pentominoes, 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

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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:
  Solutions: 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 Pentacubes     top
You can design more complicated 3-D-shapes and build them with pentominoes. 
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 Pentominoes top
If you want to play with pentominoes 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 pentominoes 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 pentominoes 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. 

Pentominoes im Internet top

German

Andrew Clarke
Polyominoes

B.Berchtold
Pentominoes - Lösung 6x10 - Applet Online

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

Thimo Rosenkranz
Pentomino-Figuren

Wikipedia
Pentomino, Polyomino

English

Andrew Clarke
Polyominoes

c.w.ricken
play pentacubes online

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

Eithan Samara
Pentominoes-3D Solver

Eric W. Weisstein (MathWorld)
Pentomino, Polyomino

Gerard's Home Page
Gerard's Universal Polyomino Solver

Kevin Gong
The Mathematics of Polyominoes

Michael Reid
Michael Reid's polyomino page

Miroslav Vicher (Miroslav Vicher's Puzzles Pages)
Polyominoes

Snaffles home page
Pentomino Relationships

Torsten Sillke
Tiling and Packing results

Wikipedia
Pentomino, Polyomino

Russian

Leonid Mochalov [PUZZLES of LEONID MOCHALOV] 
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)
(8) Pieter Torbijn: Pentominos Bridges, Cubism For Fun 59, November 2002