Introduction
Pentominoes are shapes made from five joined squares.
- Each square is joined with at least one common side.
- There are twelve ways five squares can be joined and they are these shapes.
- They have been named accordingly to their casually related letter shapes:
T, U, V, W, X, Y, Z, F, I, L, P, and N.
As a mnemonic device, to remember the names, the letters from the end of the alphabet (TUVWXYZ) and the word FLIPN for the five other letters.
There are two restriction to what counts as a pentomino.
- If a shape can be rotated to look like another, the two shapes are the same.
- If a shape can be flipped to look like another, the two shapes are the same.
History:
Pentominoes are said to have been invented by Solomon W. Golomb in 1953 when he gave a talk to the Harvard Mathematics Club.
While he was the first to use the name, pentominoes, the idea of five squares and how they might be arranged has been around since a much earlier time.
The earliest reference to a puzzle of this sort appears to be in The Canterbury Puzzles, by Henry Dudeney, published in 1907. The idea is so obvious that it may have been known to many people independently.
Challenges & puzzles
There are two ways people are challenged with pentominoes.
First, is the making or discovering the pentominoes. Which are of two types.
- The different ways five squares (12) can be arranged to make the pentominoes. Directions for Pentomino Pieces challenge
- The different ways five squares can be arranged make open box patterns (8). Directions for Pentomino Boxes challenge
Second, are area puzzles, rectangles and other shapes, to cover with the pentominoes. Harder than it sounds. Directions for area puzzles.
Challenge to find all twelve pentomine pieces
Ask. How many ways do you think five squares can be arranged to make different shapes?
Use squares to demonstrate - How each square must be joined with at least one common side. And the two restriction to what counts as a pentomino.
- If a shape can be rotated to look like another, the two shapes are the same.
- If a shape can be flipped to look like another, the two shapes are the same.
When everyone knows what a pentomino is, let them search for as many as they can find and record.
Provide a sheet of graph paper and challenge them to find all the ways five squares can be arranged (12) to make a shape.
The Solution is the pentominoes.
Box patterns challenge
Explain that five squares will make a box.
Demonstrate how to put five squares together to make the X shape and show them how it can be folded into a box (without a lid).
Challenge them to find all other possible ways that five squares can be put together to make boxes (8).
Can provide graph paper to draw box patterns and graph paper to fold.
Solution:
Pentominoes box and not box patterns.
Pentomino Puzzles
Puzzles with 60 squares for 12 pentominos
Since there are twelve shapes made from five squares, their combined area is sixty squares. Which can be a challenge to find all the different ways the shapes can be used to cover different areas with 6o squares.
Puzzle rectangles using all pentominoes can cover an area of 60 squares (5 x 12) which includes four different rectangle puzzles - that pentominoes can cover.
- 3 x 20, ( 2 solutions)
- 4 x 15, ( 368 solutions)
- 5 x 12, ( 1 010 solutions) and
- 6 x 10 (2 339 solutions).
With 2,339 solutions, you might expect finding one would be quite easy. If so, you are due for a surprise!
One 6x10 solution:
Solution sources:
The first investigation of these puzzles by computer was probably by Dana Scott in 1958. The results were summarized in an article by C. J. Bouwkamp in the Journal of Combinatorial Theory in 1969. and The Puzzling World of Polyhedral Dissections, Chapter 2 - Two-Dimensional Combinatorial Puzzles, by Stewart T. Coffin
Other puzzles
- Use the pentomino pieces to make rectangles with areas of: 35, 45, 55, 65, 85, 125.
- Create a cross with all 60 squares, that is 9 wide, 14 high, and 3 wide at its narrowest.
- Use the penomino pieces to make a rectangle 5 x 8 (40).
Create yout own puzzles!
Graph paper to make a box patterns
Draw a box pattern, then cut and fold to see if it would make a box.
