Pentominos are possibly the most popular mathemetical jigsaw puzzles.
The pentomino is a puzzle that is formed of no more than 12 pieces, which are all possible combinations of 5 squares each.
You may rotate the pieces in all directions and even turn them upside down, so asymmetrical pieces are available in mirror state.
Many surfaces can be filled with the pentomino pieces as well as 3D-shapes with 3D-forms of the pieces.
The pieces are built of equal squares, it's completely normal when combinations of pieces fit pretty well, but you have no clue whether a solution is close ... until the last piece fits in too.
Pentominos are part of the polyominos. More about Polyominos

The pieces fit on many symmetrical surfaces:

rectangle 3 x 20 - only 2 different ways to fit !

Play this puzzle yourself

rectangle 4 x 15 - 368 ways to fit the pieces


rectangle 5 x 12 - 1010 ways


rectangle 6 x 10 - 2339 possible solutions!

Play this puzzle yourself

Square 8 x 8 with a central hole of 2 x 2 - 65 ways of fitting

All solutions at 'deelder's site

Square 8 x 8 without corners - 2170 solutions


There exist many other symmetrical surfaces, like this cross (9 ways)


Download
I have written a pentomino solver in Delphi-6 for Windows, that quickly shows all solutions.
With this Delphi-application you can find the solutions of almost any puzzle surface.
Free download Pentomino solver
Question to the audience:
Which puzzle surface provides the largest number of possible solutions???

Tenyo issued 2 pentominos in his 'beat the computer' series:

No. 5 - The rectangle 6 x 10

No. 8 - The square 8 x 8 with a central hole

If the pieces are built by cubes, you're able to create many 3D-shapes.
3D Pentominos

Links to other sites:
Gerard Putter's site (dutch)
3D-pentomino on 2ndlook.nl
Broken chessboard puzzle