A pentomino is a flat figure made from five unit squares joined along an edge. In the diagram below, figures P and Q are pentominoes but figure R is not.
If you think of each pentomino as a flat shape that can be moved around the plane, then P and Q are two distinct pentominoes. There is no way that we can rotate the P shape so that it matches the Q shape.
How many distinct flat pentominoes can you find?
If you now think of each pentomino as a physical tile that you can hold in your hand, you can easily see that figures P and Q are reflections of each other – you can flip one over and it looks like the other one. These are known as ‘free’ pentominoes.
How many distinct free pentominoes can you make? (As P and Q now count as just one shape, there are fewer free pentominoes than flat ones!)
Warm-up over, let’s get down to the real question! If you are only allowed P and Q pentominoes, in how many ways can you tile the following rectangles, making sure there are no gaps, no overlaps and no overhangs?
a) 5x4 . . . . . . . . . . b) 5x8 . . . . . . . . . . c) 5x10 . . . . . . . . . .
Pentamino (for Windows). Simple to install and... simply frustrating!
If you prefer an online game - and an easier starting level - then try the Pentominoes Game at Scholastic.
Pentominoes also makes an interesting 2 or 3 player game. Taking turns, the aim is to create ‘holes’ in the grid that no pentomino pieces can fit into. However, your opponent is trying to do the same thing. The winner is the last player who has successfully placed a piece on the board.
There are numerous implementations of pentominoes as both a game and a puzzle. If you find one you particularly like, let everyone know in a comment below.
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