Get 3-d Pentomino Puzzle - Learning Resources

As every pentomino has an area of 5 square units, two pentominoes form a shape with area 10 square units. As every pentomino has an even perimeter, and joining pentominoes reduces their combined perimeter by twice the length of the overlap, every combination of pentominoes will also have an even perimeter.

Pentomino configurations and solutions. Take five identical squares. Arrange the squares so that each square shares at least one edge with at least one of the other four squares. Find all such arrangements, then remove any arrangement that is the same as any another arrangement turned or flipped in any way.

There are exactly 2339 solutions, excluding trivial variations obtained by rotation and reflection of the whole rectangle, but including rotation and reflection of a subset of pentominoes (which sometimes provides an additional solution in a simple way).

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.

There are twelve possible shapes in a set of unique pentominoes, named T, U, V, W, X, Y, Z, F, I, L, P, and N. An easy way to remember all the letters in a pentominoes set is to look at the word, FILiPiNo, and remember the end of the alphabet, TUVWXYZ.

Pentominoes (made from 5 squares) are the type of polyomino most worked with. There are only 12 in the set (because shapes which are identical by roatation or reflection are not counted). This means that they are few enough to be handleable, yet quite enough to provide diversity.

Rules: Divide the grid into pentominoes (five-cell regions) so that no two pentominoes of the same shape (including rotations/reflections) share an edge. A cell with a letter in it must be part of the pentomino shape normally associated with that letter.

Pentomino configurations and solutions. Take five identical squares. Arrange the squares so that each square shares at least one edge with at least one of the other four squares. Find all such arrangements, then remove any arrangement that is the same as any another arrangement turned or flipped in any way.

They express simplicity. The twelve pentominoes can be arranged to fill simpler shapes, namely rectangles of the area of 5 x 12 = 60 unit squares. For example, rectangles with the sides (6 x 10), (5 x 12), (4 x 15), (3 x20) can be composed of the twelve fundamental shapes.