## Pentominoes

#### Problem

A pentominoe is a shape made up of five unit square and there there are exactly twelve different pentominoes.

Arrange the pentominoes to form a rectangle measuring 10 by 6.

#### Solution

Incredibly, there are 2339 solutions to this problem. Despite this, it still remains very difficult to find a solution; here is one solution:

Extensions

• Arrange the pentominoes to form a 12 by 5 (1010 solutions), 15 by 4 (368 solutions) or a 20 by 3 rectangle (2 solutions).
• By using the set of pentominoes and an additional 22 tetrominoe, form a square measuring 8 by 8.
• It is possible to take any pentominoe and, using nine of the remaining eleven pieces, form a figure that is mathematically similar – an enlargement, scale factor 3. Try to create similar figures for each of the twelve pentominoes.

For the definitive guide to dissection problems (and solutions) I would refer you to Stewart Coffin's amazing The Puzzling World of Polyhedral Dissections; found at: http://www.johnrausch.com/PuzzlingWorld/. In particular, there is an entire section dedicated to Pentominoe problems: http://www.johnrausch.com/PuzzlingWorld/chap02d.htm.

Problem ID: 15 (Sep 2000)     Difficulty: 1 Star

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