Translation. Region: Russian Federation –
Source: Novosibirsk State University –
An important disclaimer is at the bottom of this article.
"Labyrinth"—a laboratory for intensive intellectual development—is the name of the 2025 on-site mathematical immersion program for first- to fourth-year students in research groups. Faculty of Mechanics and Mathematics of Novosibirsk State University— a regular event of the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences with the support of the International Mathematical Center. The immersion took place in late October at the O. Koshevoy health and educational camp. Nearly two dozen students passionate about mathematics solved problems, participated in creative competitions and a conference, watched films, and attended lectures.
Only four problems were assigned over the two days, equal to the number of teams. But, as usual, most of them had no clear answer, requiring careful thought to come up with an engaging solution, which each team presented at the final conference. The problem conditions can be considered a framework. Teams can refine them, change them, narrow or expand them to present a general, attractive solution. For example, the question, "Into how many squares can a 7 by 5 rectangle be divided?" quickly evolved into a search for the minimum number of squares, since everyone knows the multiplication table. The goal was to find some universal formula for dividing the squares. Many teams quickly arrived at Euclidean division with remainder, but no one came up with a hypothesis for the minimum number of squares.
The second problem required fitting the maximum number of fragments with four vertices into a graph. The most successful graph was one in which edges were drawn between all vertices, and then many more edges ("fuzzies") were added to each vertex. Lev Zhukov and Timofey Vasiliev drew attractive, "fluffy" graphs.
The third problem, at first glance, seemed simple enough. What strategy should you follow when erasing one fragment at a time in a chain so that after your move you don't end up with two segments with the same number of links, while your opponent does? A clear algorithm for a sure victory was not found. The question remained open.
"Do you know the solution to this problem?" the students asked.
"Of course not!" the organizers admitted frankly. They weren't being disingenuous. Solving problems with predetermined answers, like a test in school, is a bit boring for real researchers, whether they're in their first or fourth year. The students laughed, of course, but they were pleased to be on equal terms with doctors and candidates of science. Many of their eyes lit up: "These are real problems!"
Klim Bagryantsev offered a beautiful and colorful image (but not a solution) of a problem where one had to divide a rectangle into four pieces of a "triomino" game. This is a well-known type of problem involving filling a surface with identical fragments. The result is a fractal canvas of four colors, similar to a Sierpinski triangle, where small shapes compose similar larger ones.
Each team had its own name and mascot. The name "Outegral" clearly aspired to a new concept, the opposite of "integral." The "Nail Rinatovich" team was named after a classmate who was absent from the team and whom the students clearly missed greatly. The "Koala" team's full name was "Koala Eucalyptus," and the "Mathematini" team's mascot was a true artistic masterpiece, adorning the event until its departure.
"This year's immersion was a very emotional experience for me," said fourth-year student Daria Koroleva. "I'm so glad the first-year students managed to keep their cool and solve the problems. It was a lot of fun with them. I really enjoyed the problems. I was a little disappointed that I couldn't find a perfect solution, but that's not always possible, but solving them was a lot of fun. You get completely immersed in the research process, generating hypotheses, considering different examples, proving or finding counterexamples, communicating, and discussing ideas. Researching problems with other students is a unique experience."
In addition to math problems, there were night photography contests for the most geometric and the scariest shadow. Some scenarios required participants to perform complex choreography and even some acrobatic skills, while others required only a little mischief and ingenuity.
The country camp became a magnet not only for students and teachers—the first lecture, dedicated to mathematical billiards and Birkhoff's algebraic conjecture, was given by Andrei Mironov, Director of the Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, and Corresponding Member of the Russian Academy of Sciences.
There are a huge number of very beautiful theorems concerning Birkhoff billiards, and many still unproven conjectures. For example, the question of the periodic trajectory of a billiard ball within any convex figure. In an acute triangle, it will be periodic, meaning the ball will repeatedly hit the same points. But within a circle, there can be various periodic trajectories of a billiard ball—from an equilateral triangle to a square, a five- or even a six-pointed star. Andrei Mironov gave students a brief history of scientific research into Birkhoff's conjecture and presented several solutions, authored byoneof which in 2019 was the lecturer himself, together with a colleague from Tel Aviv University, Professor Mikhail Bial.
Birkhoff's conjecture states that every integrable billiard table is an ellipse. An elliptical curve that touches all segments of the billiard ball's trajectory, bending around them, is called a caustic. This term came to mathematics from optics, where it described the geometry of reflection and refraction of light beams such that in certain places the light gathers into particularly bright spots, for example, on the surface of the sea or inside a faceted diamond. A question from the audience asked whether caustics can intersect. The speaker replied that no one had yet solved this mystery.
Andrei Mironov spoke about his acquaintance with Mikhail Byaly at a conference in Scotland, and the organizers of the mathematical immersion recalled that a team of schoolchildren from Scotland (including children of NSU graduates) once won their annual autumn math marathon. The bizarre and vibrant, caustic intersections of scientific trajectories demonstrate that mathematics is not self-sufficient and cannot develop in a hermetically sealed manner within a single country, city, or institute. Mathematics is one. People living on opposite sides of the globe simultaneously ponder the same problems and ask the same questions.
Material prepared by: Maria Rogovaya, press service of the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences
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