Polyominoes

If we have guests, it is not unusual for them to notice my little collection of wooden puzzles on the dresser. It’s not unusual for a guest to attempt each of them in turn, and for me to spend the next couple of nights after they leave trying to put the buggers back again.

They range from the simple to the bastardly. Two of my favourites I am sure you will be familiar with, not least from Martin Gardner’s excellent series of Scientific American articles.

The Soma Cube

The 6 non-linear combinations of four cubes, when joined by the single irregular (non-linear) combination of three cubes, make a pleasing puzzle: they can be arranged to make a 3x3x3 cube. Piet Hein apparently constructed the set and the puzzle during a lecture on quantum mechanics by Heisenberg. (Click for larger versions of pictures)

Rather like Tangrams [http://en.wikipedia.org/wiki/Tangram], they can also be used to make other shapes: http://www.fam-bundgaard.dk/SOMA/FIGURES/FIGURES.HTM

More gen: http://en.wikipedia.org/wiki/Soma_cube

Pentominoes.

The 12 combinations of five squares are called pentominoes. A mnemonic to reconstruct them is “FILiPiNo TUVWXYZ”, where each capital letter calls to mind the shape of one of the targets.

They also have some pleasing puzzles associated with them, like…

  • making a rectangle 5×12 or 6×10 (testing!) or 3×20 (evil!)
  • making a circle where each pentomino differs from its neighbours by the position of a single square

More are to be found at the excellent web site of my former colleagues at the Centre for Innovation in Mathematics Teaching: http://www.cimt.plymouth.ac.uk/resources/puzzles/pentoes/pentoint.htm

I particularly enjoy the space-filling problems (jigsaws). There is even a chessboard-related puzzle, and a chessboard-based game.  Gardner explains:

What is the minimum number of pentominoes that can be placed on a checkerboard in such a way that it is impossible to place any of the remaining pentominoes on the board? This intriguing question is asked by Golomb, and he says the answer is five. Figure 79 shows one such configuration. This problem suggested to Golomb a fascinating competitive game that can be played on a checkerboard with large cardboard pentominoes cut to fit accurately over the board’s squares. (The reader is invited to make such a set, not only to enjoy the game, but also to solve pentomino problems and create new ones.)

Two or more players take turns in choosing a single pentomino and placing it wherever they wish on the board. The pieces have no “top” or “bottom” faces. As in all problems mentioned in this article, asymmetrical pieces may be used with either side up. The first player who is unable to place a piece is the loser.

Now, there is further niftiness to be had: if you construct a set from cubes rather than squares (pentacubes):

You have some 3-D jigsaw puzzles. The most satisfying is to construct a 3x4x5 cuboid. There are loads of solutions, but it could take you a while… Wikipedia will rescue you if you give up: http://en.wikipedia.org/wiki/Pentomino

Anyhow, if you find you are on top of all those puzzles, you might be interested to know that I have just acquired a Bedlam Cube (now marketed as Crazee Cube). Handle with care! It’s a 4x4x4 cube made up of a selection of 12 pentacubes (including the W, X and F pentominoes) and a single tetra-cube. http://en.wikipedia.org/wiki/Bedlam_cube

I have dissected it slowly and recorded the position of each piece, so that when I come to tackle it I know I can get it back again. [Of course, having done that, I find someone has an online Bedlam Cube solver which taunts you with a random display of one of the 19,000+ solutions… http://www.danieltebbutt.com/bedlam.html%5D

D

P.S. http://cheesybug.co.uk/shop/article_Retrobed/Retro-Bedlam-Cube-%28Yellow,-Red-and-Blue%29.html suggests ’36 months and up’ for the Bedlam Cube, which is my current estimate for solving it independently…

P.P.S. Interested in obtaining cubes of different colours? Try http://www.boardgameextras.co.uk/index.php?cat=12

P.P.P.S I think in fact I have a pretty poor ‘spatial intelligence’ – Sally is the one for that

P.P.P.P.S Hat-tip: Nick Givens, for suggesting how to draw the pieces

Advertisements

Leave a comment

Filed under Uncategorized

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s