Alex Bellos's Monday puzzleMathematics This article is more than 3 years oldCan you solve it? Are you a master of reflection?
This article is more than 3 years oldA colourful puzzle for squares
UPDATE: The solutions can be read here.
Felt tips at the ready! Today’s puzzle involves colouring in.
The image below shows a square divided into eight segments, and the four ‘mirror lines’ of that square. In other words, when you reflect the square across each of these axes, the square looks exactly the same.
The point of today’s puzzle to see how the mirror symmetries change when we start colouring in the segments. For example, if we colour the top two segments green, as below, and leave the rest white, the square now only has a single mirror line, the vertical one:
On to the challenge:
How many different ways can you colour the square with two colours such that
1. The only mirror line is the horizontal mirror line?
2. The only mirror line is the right-sloping diagonal mirror?
3. The square has both horizontal and vertical mirror lines, but no diagonal mirror lines?
4. The square has a diagonal mirror line and a horizontal mirror line, but not a vertical mirror line?
The question only concerns a square divided into 8 segments like the ones above. But be very careful! For example, this square is not a solution for 1 since both the horizontal and vertical mirror lines are mirror lines.
Also, be sure not to double-up on cases. For example, the two colourings below are considered the same. (We are not concerned with green and white, just between two areas of different colours.)
If you want some neatly drawn squares to doodle on, there’s a printable page of squares here.
Today’s colouring-in conundrum comes from a lovely new book, Beautiful Symmetry: A Coloring Book About Math by computer scientist Alex Berke, a former Google software engineer who now works at MIT. (Full disclosure: I liked the book so much I wrote the introduction.) Her idea is to introduce mathematical ideas about symmetry through the process of colouring in. It’s an engaging introduction to the maths of ‘group theory’ as it explores ideas such as cyclic groups, frieze groups and wallpaper groups in a playful and very accessible way.
I’ll be back at 5pm with the solutions.
Meanwhile, NO SPOILERS! Please talk about felt-tip pens.
UPDATE: The solutions can be read here.
Thanks to Alex Berke for today’s puzzle.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
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