Cleave Books
Space-filling Problems 3

12 pentominoes (each made of 5 squares) must cover between them, a total of 12 x 5 = 60 squares.
One of the ways this can be done is by filling a rectangle measuring 3 by 20.
Such a rectangle, with just 3 pentominoes placed in it is shown below. It is the start of a solution.

pentomino 3x20
16. Find the complete solution to filling the 3 by 20 rectangle.

A rectangle of area 40 squares can be filled with 8 pentominoes.
One way of doing this is in a 5 by 8 rectangle and one is shown on the right.
pentomino 5x8
17. Find a way of covering the same area but in a 4 by 10 rectangle.
18. Cover an area of 45 squares in a 5 by 9 rectangle.
19. Fill a rectangle of area 50 squares.
20. Fill a rectangle of area 55 squares.
Using all 12 pentominoes to fill a rectangle can be done in 3 different ways.
As a 3 by 20 rectangle; as a 5 by 12 rectangle; as a 6 by 10 rectangle.
The 3 by 20 was done in problem #16 (above).
One solution to the 5 by 12 (from among several thousand) is shown below.
pentomino 5x12
21. Find a different way to fill the 5 by 12 rectangle.
22. Use the full set of 12 pentominoes to fill a 6 by 10 rectangle.

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