Earlier this month, Russia’s Prime Minister Mikhail Mishustin marked the first year of the school year by attending a sixth-grade math class at one of his country’s top science-oriented schools.
The class studied a problem about business. “Why do you need to do business projects in [school]? ” he asked. “Basic knowledge is needed here, right?”
He then wrote the following problem on the board.
Construct a perpendicular from the (red) point of the circle to the diameter, without using any measuring devices.
In other words, with a circle with a diameter marked on it and a point on the circle, you can 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 issue is the seemingly outrageous limitation of not allowing measuring instruments, which means you can not use a compass or a marked ruler. All you get is an unmarked ruler to draw straight lines.
The photo of Mishustin at the top of this article shows you how to get started. (He shows that the angle subtlied by a point on a circle to the two ends of a diameter is a right angle.) Some “basic knowledge” of triangles may be useful. For example altitude in a triangle, the line is from a corner that meets the opposite side at a right angle. In acute triangles, the three heights will always cross.
I chose this question today because it is a wonderful puzzle, and also because I thought it was curious that one of Russia’s most powerful politicians used a geometry question as an advertising stunt. It is not every day that you see a politician print a math problem on a board and solve it, no matter what country they come from.
Mishustin, a trained engineer, told the Sixth Forms: “It seems like at your age it would be good to get some basic things. And when you have mathematical knowledge, physics, chemistry, you will be able to solve all problems, including business. ”
I will be back 17.00 UK with the solution. PLEASE NO SPOILERS.
Thanks to the International Congress of Mathematicians (ICM) 2022 for using the illustrations. ICM 2022 will be held in St. Petersburg next year.
The Russian government has been criticized by the international mathematical community for what it considers to be human rights abuses against mathematician and anarchist Azat Miftakhov, who was sentenced to six years in a penal colony earlier this year for alleged hooliganism. A request for his release has been signed by more than 3,000 academic mathematicians around the world.