# Can you solve it? Are you smarter than Britain’s teenage brainiacs?

Today’s puzzle celebrates the UK’s outstanding performance at last week’s European Girls’ Mathematical Olympiad, which is the world’s most prestigious female-only maths competition for pre-university students.

Yuhka Machino and Jenni Voon, both aged 17, placed 6th and 7th overall, each earning gold medals. Overall, the UK ranked fifth out of 55 countries, behind Russia, the USA, Peru and Romania. (The event was held online, allowing 17 non-European countries to participate as guests.)

The following puzzle is taken from the Mathematical Olympiad for Girls, a school challenge that helps select the UK team for the European olympiad.

Painting the houses

Each of 100 houses in a row are to be painted white or yellow. The residents are quite particular and request that no three neighbouring houses are all the same colour.

(a) Explain why no more than 67 houses can be painted yellow.

(b) In how many different ways may the houses be painted if exactly 67 are painted yellow?

Olympiad questions like this one are designed to reward mathematical nous, rather than proficiency in any techniques that you may have learned at school. The proof is not complicated if you approach the problem in the right way.

If you fancy a tougher challenge, here’s the first problem from last week’s European Girls’ Mathematical Olympiad.

Fantabulous numbers

The number 2021 is fantabulous. For any positive integer m, if any element of the set {m, 2m + 1, 3m} is fantabulous, then all the elements are fantabulous. Does it follow that the number 20212021 is fantabulous?

All four members of the UK’s team – Machino, from Millfield School, Voon, from Landau Forte College, Aanya Goyal, from Alleyn’s School, and Eleanor MacGillivray, from King’s Ely – got perfect scores for this question.

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I’ll be back with the solutions at 5pm UK time today. PLEASE NO SPOILERS.

Mathematical Olympiads have been around for decades but are dominated by boys, hence the decision in 2012 to start a Europe-wide competition for girls only, with the hope that it might increase female participation in maths and science olympiads in general. (At the 2020 International Mathematical Olympiad, less than 10 per cent of contestants were girls. Machino was the only girl among the 50 contestants who won a gold medal.)

Disclaimer: in preparation for the inevitable snipes about the headline, no I am not saying that mathematical ability is the only measure of smartness in the world. It’s just a headline!

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 puzzles, most recently the Language Lover’s Puzzle Book.

Thanks to the brilliant UK Mathematics Trust, which promotes maths in UK schools via national competitions and other resources, for today’s puzzle. You find out more about the British Mathematical Olympiad and the Mathematical Olympiad for Girls on this webpage.