Today’s puzzles come from Alain Nicolas, a Frenchman who has been called the world’s finest artist of Escher-style tilings. (That’s to say, tilings with no gaps or overlaps in which each tile is in the shape of a living creature, as pioneered in the 1930s and 1940s by the Dutch artist MC Escher).
In each puzzle, you will be presented with an outline, such as the one below left. The challenge is to draw a line that divides it into a certain number of pieces of identical size and shape. In this case, the solution, with two pieces, is below right.
The two pieces have the shape of a fish.
Here are the puzzles. You can click here for a printable page or use a drawing app like Markup, free inside Photos on your iPhone.
Puzzles that require you to divide an outline into two or more congruent shapes are a standard exercise to develop geometric intuition. In all of today’s puzzles we get the extra pleasure of seeing the outline of an animal emerge.
The fish shape in the example above – as well as the shapes that emerge in the other puzzles – are all shapes that “tessellate”, meaning they can be arranged to cover a flat surface leaving no gaps or overlaps. Here’s a tessellation of the fish:
Alain Nicolas, aged 73, was inspired to create his own tessellations on seeing the work of Escher four decades ago. Escher’s tessellations of interlocking birds, fish and lizards are some of the most recognisable mathematical art of the twentieth century; striking and playful as well as breathtakingly ingenious.
Nicolas’ work is also stunning and witty. Now retired, he spends half his free time designing tessellations and recently finished his 400th. You can see many of them on his extensive website (but don’t peek until you have solved the puzzles!).
Drawing tessellations is not easy. It takes a lot of geometrical acuity to make shapes that fit together and are convincing representations.
David Bailey, a British tessellation artist, believes that Nicolas is the best tessellation artist in the world. “His work has everything, recognisable silhouettes, quality, variety, number, level of innovation, next to no padding, and all rendered to a most pleasing standard of finish. Bravo, Alain!”
Nicolas has – like Escher – no background in maths, but says all that is required is a sense of wonder and a desire to always do better. Here is a self portrait, sitting in a bar, reading his own book, and calling the waiter with his finger.
I will post the solutions to the puzzles at 5pm UK time today.
There is a huge amount of information about tessellations on the web. These two are a good place to start:
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.
I’m the author of several books of popular maths, and the Football School book series for kids that explains the world through football.