Pentominoes

Pentominoes is the set of shapes created by combining 5 squares or cubes, with faces touching each other, in every possible combination.  Mirror images and rotations do not count as additional combinations.

With this, we achieve 12 unique shapes, as seen below:

p111.gif (5283 bytes)

They tend to resemble the following letters, respectively:

F  I  L  N  P  T  U  V  W  X  Y   Z

 

What can you do with these 12 puzzle pieces?  You would be amazed!

(Remember, each piece contains 5 squares, so there are 60 cubes total.)

Using all 12 pieces, you can make:

  1. A 6 x 10 rectangle (see photo below)
  2. A 5 x 12 rectangle
  3. A 4 x 15 rectangle
  4. A 3 x 20 rectangle
  5. An 8 x 8 square with 4 pieces missing in the middle (see photo below)
  6. An 8 x 8 square with 4 pieces missing in the corners
  7. An 8 x 8 square with 4 pieces missing almost anywhere
  8. A 3 x 4 x 5 cube (see photo below)
  9. A 2 x 5 x 6 cube
  10. A 2 x 3 x 10 cube
  11. A 2D replica of each piece, only three times larger
  12. A 5 x 13 rectangle with the shape of 1 pentomino piece missing in the middle
  13. Shapes with jagged edges
  14. Tessellations using a pentomino
  15. Hundreds of other shapes!

 

You can make your own set of pentominoes!

Run, don't walk, to the nearest craft shop and buy 60 wooden cubes, each measuring about 3/4" on a side.  Glue sets of 5 together into each of the 12 pieces shown above.  (Make sure the U isn't too tight for the others to fit into it!)  Then, have hours of fun trying to create each of the above shapes.

pent2.jpg (2796 bytes) pent5.jpg (2218 bytes) pent6.jpg (2989 bytes) pentsml.jpg (2846 bytes)

The photo on the right is a pentominoes set I purchased in Shanghai, China.  Each cube measures about 15mm on a side.  It's a very nice set, great size, clever box with lid, and only cost about 120RMB ($15).  The manufacturer is Dr. Br@in. 

If you'd like to check out some other useful links on Pentominoes, try these:

http://www.johnrausch.com/PuzzlingWorld/   A fantastic site on several types of puzzles.

http://www.johnrausch.com/PuzzlingWorld/chap02d.htm   This takes you right to the section on pentominoes.
Has all solutions to the rectangles above (items 1,2,3,4).

http://www.xs4all.nl/~gp/pentomino.html   Nice review.

http://www.xs4all.nl/~gp/PolyominoSolver/Polyomino.html   Provides solvers for several configurations.

http://www.cut-the-knot.com/  This doesn't have any pentominoes that I could find, but has lots of great info on math!

http://www.wins.uva.nl/misc/pythagoras/polyominoes.html   More great links.  There is a lot out there.

http://godel.hws.edu/java/pent1.html   Wonderful solver for the 8 x 8 square with four 1 x 1 holes.

http://www.pentomino.be.tf/   A nice student site with additional puzzle links.

 

For books on pentominoes, I highly recommend the following:

Mathematical Puzzles & Diversions
Martin Gardner
LCCN 59-9501
Simon and Schuster
NY 1959

Creative Puzzles of the World
Pieter van Delft and Jack Botermans
LCCN 77-80234
ISBN 0-8109-2152-9 (pbk)
ISBN 0-8109-0765-8
Harry N. Abrams, Inc.
NY 1978

Polyominoes
Solomon W. Golomb
ISBN 0-691-02444-8 (pbk)
ISBN 0-691-08573-0
Princeton University Press
Princeton, NJ 1994

Pentominoes
Jon Millington
ISBN 0-906212-57-X (pbk)
Tarquin Publications
Norfolk, England 1995

Tilings and Patterns
B. Brunbaum and G.C. Shephard
ISBN 0-7167-1193-1
W.H. Freeman and Co.
NY 1987

For more selections, check out amazon.com for other books by Martin Gardner, pentominoes, puzzles, etc.

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