Today’s puzzles are of a piece. Two, three, and five pieces, to be exact.
In each problem you are presented with a set of non-symmetrical shapes. The challenge is to rearrange them without overlaps so the combined shape has a line of symmetry.
1. Triangle 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.
2. Tetromino triplets
This one for the tetris fans. Here are three L-shaped tetrominos (the technical term for a shape made from four squares joined along grid lines.)
Can you rearrange them with no overlaps 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 had cut out the shapes. There is one solution just by sliding the shapes around, and one way in you have to pick one shape up and flip it around before putting it back on the table.)
3. Triomino quintuplets
Same again, this time with five L-triominos (i.e a shape made from three squares.) Can you rearrange them with no overlaps so that the combined shape has a line of mirror symmetry?
UPDATE: 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 the individual triomino does not count. That would be too easy.)
The most entertaining (and helfpul) way to solve these problems is to cut them out of paper and do the rearranging by hand.
However, if anyone wants to make an interactive version, I’m sure all fellow readers would appreciate it. Put the link in the comments below, and I’ll add it here.
UPDATE: INTERACTIVE VERSION HERE Thanks to User0
I’ll be back at 5pm UK with the solutions. PLEASE NO SPOILERS. Instead discuss early computer games, or your favourite tetrominoes.
UPDATE: Read the answers here.
Today’s puzzles were devised by Donald Bell, former director of the National Engineering Laboratory. If you would like to hear more about his passion for polyominoes, here’s a talk he gave about them.
I’ve been setting a puzzle here on alternate Mondays since 2015. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
