Earlier this month, Russia’s Prime Minister, Mikhail Mishustin, marked the first day of the school year by visiting a sixth form maths class at one of his country’s top science-oriented schools.
The class was studying a problem about business. “Why do you guys need to do business projects in [school]?” he asked. “Fundamental knowledge is needed here, right?”
He then wrote the following problem on the blackboard.
Construct a perpendicular from the (red) point on the circle to the diameter, without using any measuring devices.
In other words, given a circle with a diameter marked on it, and a point on the circle, can you find a way to draw a line from the point that hits the diameter at a right angle. (As marked in green above.)
The beauty of this question is the seemingly outrageous restriction not to allow measuring devices, which means that you cannot use a compass or a marked ruler. All you are allowed is an unmarked ruler to draw straight lines.
The photograph of Mishustin at the top of this article shows how you might begin. (He is showing that the angle subtended by a point on a circle to the two ends of a diameter is a right angle.) Certain ‘fundamental knowledge’ about triangles may be useful. For example, the altitude of a triangle is the line from a corner that meets the opposing side at a right angle. In acute triangles the three altitudes will always intersect.
I chose this question today because it’s a lovely puzzle, and also because I found it curious that one of Russia’s most powerful politicians used a geometry question as a publicity stunt. It’s not every day you see a politician write out a maths problem on a blackboard and solve it, whatever country they might be from.
Mishustin, who is an engineer by training, told the sixth formers: “It seems to me that at your age it would be good to get some fundamentals. And when you have mathematical knowledge, physics, chemistry, you will be able to solve any problems, including business.”
I’ll be back at 5pm UK with the solution. PLEASE NO SPOILERS.
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. I also give school talks about maths and puzzles (restrictions allowing). If your school is interested please get in touch.
Thanks to the International Congress of Mathematicians (ICM) 2022 for use of the illustrations. The ICM 2022 will be held in Saint Petersburg next year.
The Russian government has come under criticism from the international mathematical community for what they consider human rights abuses against the mathematician and anarchist Azat Miftakhov, who was sentenced to six years in a penal colony earlier this year for alleged hooliganism. A petition for his release has been signed by more than 3000 academic mathematicians around the world.