## On the tiles

Submitted by Marianne on December 8, 2017Our colleague Becky Warren, who runs the Hands On Maths Roadshow, has been distracting us with the wonderful puzzles she has been making, both virtually for the Millennium Mathematics Project and also as beautiful hand-made versions to play with in real life.

Here's one example that is particularly satisfying to try for yourself. Start with a dodecagon (you can see how to construct one here, or just print out your own version from this PDF) — can you work out the area of this shape?

This might not seem obvious, but there is a clever way of working it out if you first cut the dodecagon up into triangles. The triangles can also be rearranged to form three squares, which quickly lead you to discover the answer, providing a visual proof for a mathematical questions. This kind of problem, where you can rearrange pieces of a puzzle in two ways, is one of Becky's favourite type of puzzles.

Another favourite puzzle of ours involves tiling the plane with pentagons. Impossible you say? You're right, you can't tile the plane with regular pentagons, as we've seen here. But in 1918 mathematicians discovered that you can tile your bathroom floor with certain irregular pentagons whose sides aren't all the same length. People have been hunting for all the different types of pentagons that can do this, and just two years ago mathematicians discovered the first new tile in 30 years.

New mathematical discoveries like this often inspire Becky, and investigating these tilings with pentagons led her to a new favourite tiling called the Bow Tie, which you can play with on our sister site NRICH or make yourself out of paper. You can arrange the Bow Tie tiles in lots of ways, some have symmetry, some have none, some have different shaped holes, and some neatly fit together to give you a novel tiling idea for your bathroom floor.

A bow tie tiling.

"To construct a puzzle I start with a mathematical idea," says Becky. "At the moment I'm focussed on tessellations. The recent news about pentagons led me to find this particular pentagon — the Bow Tie. It's [one of the ones already known to tile the plane] but I thought it was a particularly interesting one."

There's another recent discovery that has caught Becky's interest. Last year there were breakthroughs in *monohedral disc tilings*: circles that are cut into equally shaped pieces. "Most people think of it as cutting up a pizza so that all the pieces are the same, but you can also do it with different shapes," says Becky. Again, these tiles can be arranged in many different ways, which led Becky to create some really beautiful sets of tiles you can play with in real life.

"That's the idea — taking some nice bit of maths, perhaps something that has come up recently, and seeing if you can turn it into a puzzle."

So if you would like some ideas for Christmas gifts you can make yourself, why not check out some of these activities on our sister site NRICH:

- L-triominoes
- Four triangles puzzle
- Putting two and two together
- Repeating patterns
- Making rectangles, making squares
- Equal equilateral triangles.

But if you'd like someone else to do the hard work for you, you can always visit Becky's Etsy shop.

One of Becky's wooden puzzles. The tiles can be arranged in many different ways. You can see (and buy) more here.