Earlier today, I posed three questions from a Russian math competition that was used to promote the International Congress of Mathematicians, which will take place in July next year in St Petersburg.

Table of Contents

## Pet Swap

Imagine a scenario where a cat is perched on a table while a tortoise crawls on the floor right beneath it. The distance from the cat’s ears to the top of the tortoise’s shell measures 170 cm. Now, let’s switch the positions of the pets. As a result, the distance from the cat’s ears to the top of the tortoise’s shell is reduced to 130 cm. The question arises: what is the height of the table?

The solution is actually quite simple. We can take the table from the first picture and place it on top of the table from the second picture, aligning the tortoises. By doing this, we create a scenario where the distance from the ears of the bottom cat to the ears of the top cat is 130 cm + 170 cm, which equals 300 cm. Interestingly, this distance is equivalent to twice the height of the table, meaning the answer is 150 cm.

## Circular Thinking

Let’s consider Volodya, who is running around a circular track at a constant speed. On this track, two yellow marks are strategically placed. When Volodya begins his run, he finds himself closer to the first yellow mark for 2 minutes. He then switches and becomes closer to the second yellow mark for 3 minutes before eventually returning closer to the first yellow mark once again.

The question is: how long does it take for Volodya to complete one full circle?

To solve this puzzle, let’s label the first and second yellow marks as A and B, respectively. We can then mark points C and D on the track, creating a diameter so that every point on CD is equidistant from A and B. This essentially divides the circle into two halves, with one half closer to A and the other closer to B.

Since Volodya spends 3 minutes closer to B, we can conclude that he runs the CBD arc in 3 minutes. Consequently, he completes the entire circle in 6 minutes.

It’s worth noting that the fact Volodya was closer to the first mark for the first two minutes doesn’t affect the time it takes for him to run a full circle.

## Path of Glory

Now, let’s consider Olga, who must navigate a 3×3 grid by moving either horizontally or vertically from cell to cell. Olga’s objective is to write down the digits she encounters during her path, which should result in the number 84937561.

The question is, what is the largest number Olga can possibly write down during her journey through the grid?

To tackle this challenge, let’s begin by recognizing that a nine-digit number is inherently larger than a number with fewer digits. Hence, we should focus on finding the answer among the possible nine-digit numbers. Additionally, it’s evident that the highest number will be one that starts with the highest digit.

If we color the squares in a chessboard pattern, as shown on the left side, we can observe that the color of the cells should alternate as Olga walks along her chosen path. Since we have five white cells and four black cells, it becomes clear that the journey must commence on a white cell. The largest number present in the white cells is 5.

At each step, Olga needs to maximize the next digit position by moving to the cell with the highest possible digit. Following this principle, the numbers 7 and 3 are determined. Next, when considering the neighbors of number 3, we find that 9 is the maximum number. However, proceeding to cell 9 would lead to an incomplete path, causing the square to split into two disconnected areas. Similar constraints prevent Olga from moving to 8 from 3. Consequently, the only viable option is to proceed to 6, leading us to the following chain: 573618492.

I would like to express my gratitude to the ICM 2022 for allowing me to utilize these intriguing puzzles. For more information about the event, kindly visit the 5 WS website. Sources: 1: Maths clubs folklore. 2. Moscow Mathematics Olympiad 2015 3. Math Fest 2012.

Remember, I post a new puzzle here every two weeks on a Monday. I’m always on the lookout for great puzzles, so if you have any suggestions, feel free to email me. Additionally, I offer school talks about math and puzzles, subject to any applicable restrictions. If your school is interested, please get in touch. Lastly, don’t forget that I’ve authored several books on math and puzzles, including the recently published The Language Lover’s Puzzle Book, which would make an excellent Christmas gift!