Discover Boggler


	Pentominoes For the April 2001 Boggler in Discover Magazine By Scott Kim Please do not reproduce or distribute without permission.

Pentominoes are the twelve different shapes that can be formed by joining five squares edge to edge, as shown below. In Solomon Golomb's game of Pentominoes, two players alternately place pentominoes from a complete set of twelve pieces so they cover whole squares of a chessboard. The first player who cannot place a piece without overlapping another piece loses.

A key scene in 2001 features the intelligent computer HAL playing chess with astronaut Dave Bowman. The scene was also filmed with Hal playing Pentominoes. That scene was cut from the movie, but 2001 author Arthur C. Clarke so enjoyed pentominoes that he worked them into a later novel Imperial Earth.

1. Every Pentominoes game must end in no more than twelve moves, since there are only twelve different pentomino shapes. The shortest possible Pentominoes game on an 8 by 8 board ends in just five moves. Here is the position after the first four moves. Can you play just one more piece so the other player cannot move? Remember, each piece can only be played once.

2. Can you place three pentominoes on a 6 by 6 board so no more pentominoes can be placed? Can you find another solution that uses three completely different pentominoes?

3. Can you place four pentominoes on a 7 by 7 board so no more pentominoes can be placed? Can you find a solution in which none of the pentominoes touches the edge of the board?

4. Can you place six pentominoes on a 9 by 9 board so no more pentominoes can be placed? What about seven pentominoes on a 10 by 10 board? Hint: these boards require a fundamentally different approach than for the other problems.

Answers.

Here are the complete results known so far. Click on an entry to see what that solution looks like. If you find any improvements, please let me know and I'll add it to this page and give you credit.

Columns

Rows 3 4 5 6 7 8 9 10 11 12 13 14 15

2 1 2 3

3 1 2

4 2 2

5 2 3 3 4

6 3 4

7 4 5

8 5 6 6

9 6 7

10 7

11 9

12

13 11

14

15

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