Cleave Books
Pentominoes - A Miscellany of Problems

24. Of the 12 pentominoes, which ones could be used as the net for making an open-topped box?

25. Find the smallest number of Y-pentominoes needed to fill a rectangle completely.
(It is less than 20.)

Make each of these shapes with a full set of 12 pentominoes.
Note that 27 to 30 have (black) holes in them.
26. pentomino cross 1 27. pentomino cross 2
28. pentomino steps 29. pentomino triangle
30. pentomino grid

Solid pentominoes
If the pentominoes are made using cubes instead of squares (so that they have a 'thickness' of 1 cube) then it becomes possible to work with 3-dimensional shapes.
The simplest problem then is to put all 12 together to make a cuboid. Clearly it will have a volume of 60 cubes.
It can be done as
a 3 by 4 by 5; or a 2 by 5 by 6; or a 2 by 3 by 10.
All of these are possible.

8 of the solid pentominoes can be assembled to make a twice-size representation of most of them, but it is NOT possible in the case the I, T, W and X.

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