Earlier today I set up three symmetry puzzles for you. Here they are again with solutions.
The most entertaining way to solve these problems is to cut the pieces out of paper and do the rearrangement by hand. However, a generous reader made ‘niinteractive version available here.
1. Triangular twins
An easy one to start. These two ’30-60-90′ triangles share a side length.
(Each triangle has internal angles of 30, 60 and 90 degrees: what you would get if you cut an equilateral triangle in half.)
How would you rearrange the two triangles without overlaps to get a shape with mirror symmetry, that is, one in which a line divides the shape into two halves, one half the reflection of the other.
Find BOTH solutions.
Solution
2. Tetromino triplets
This one for the tetris fans. Here are three L-shaped tetrominos (the technical term for a shape made of four squares connected along grid lines.)
Can you rearrange them without overlapping so that the combined shape has a line of mirror symmetry?
There is one way to do it without flipping, and one way with flipping. (Imagine you have cut out the shapes. There is one solution just by moving the shapes around, and one way in you have to pick up one shape and turn it over before putting it back on the table.)
Solution
3. Triomino quintuplets
The same again, this time with five L-triominoes (ie a shape made of three squares.) Can you rearrange them without overlapping so that the combined shape has a line of mirror symmetry?
Find a solution where the line of symmetry is either parallel to, or perpendicular to, all the edges of all the triominoes. (So using the line of symmetry of an individual triomino does not count.)
Solution
Thanks to Donald Bell for today’s puzzles. Donald is a former director of the National Engineering Laboratory. If you want to hear more about his passion for polyominoes, here is a speech he gave about them.
I hope you enjoyed today’s puzzles. I’ll be back in two weeks.
I’ve been doing a puzzle here on alternate Mondays since 2015. I’m always on the lookout for great puzzles. If you want to suggest one, email me.
