Can you solve it? From Russia with logic

The largest and most important event in the mathematical calendar will take place next July in St Petersburg. The International Congress of Mathematicians is a quadrennial gathering at which many of the subject’s most prominent thinkers give lectures and the winners of maths’ most prestigious prize, the Fields Medal, are announced.

As part of the build-up to the event – which is expected to attract about 5,000 mathematicians from all over the world – the organisers earlier this year ran an online maths competition for students and the general public. Below are the first three questions from this test.

Russia has long been a world leader in mathematics outreach, having pioneered the idea of maths circles (informal maths clubs where students often try to solve problems in groups) and competitive maths olympiads.

Today’s three puzzles are all brilliant examples of the Russian style: they test logical reasoning rather than taught mathematical skills. The first one, in particular, is a classic.

1. Pet swap

A cat is sitting on the table and a tortoise is crawling on the floor directly beneath it. The distance from the cat’s ears to the top of the tortoise’s shell is 170 cm. Alena switched her pets around. Now the distance from the cat’s ears to the top of the tortoise’s shell is 130 cm. What is the height of the table?

2. Circular thinking

Volodya is running around a circular track at a constant speed. At two of the points on the track are two yellow marks. When he started his run, Volodya was closer to the first yellow mark for 2 minutes, then he was closer to the second yellow mark for 3 minutes, and then he was closer to the first yellow mark again.

How long does it take Volodya to run one full circle?

3. Path of glory

Olga must walk through the 3×3 grid below left, moving either horizontally or vertically from cell to cell, and not entering any cell more than once. Below right shows one possible path. If Olga writes down the digits she passes in order during this path, she would get the number 84937561.

What is the largest number possible it is possible for Olga to write down during a path through the grid?

I’ll be back at 5pm UK with the solutions. Meanwhile, NO SPOILERS.

Thanks to the ICM 2022 for letting me use the puzzles. You can find more about the event here. Sources: 1: Maths clubs folklore. 2. Moscow Mathematics Olympiad 2015 3. Math Fest 2012. I rephrased the questions for clarity.

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 also give school talks about maths and puzzles (restrictions allowing). If your school is interested please get in touch.

I’m the author of several books of maths and puzzles, most recently The Language Lover’s Puzzle Book, all of which would make fantastic Christmas gifts!