The science and touristic camp for the laureates of the Kangaroo competition took place from July 30 to August 8, 2024, at the Dafne II center in Zakopane. Participants from eight countries attended: China (for the first time), France, Lithuania, Germany, Poland, the United States, Turkey, and Ukraine. The official language of the camp was English.

Participants took part in mathematical workshops and competitions, hikes in the mountains, tourist excursions to Krakow and Wieliczka, and sports competitions. The highlight of the camp, following the tradition of previous years, was the Young Talents Contest.

Mathematics

Most mathematics sessions were divided into two groups: the younger group A (Cadet category) and the older group B (Junior and Student categories). Participants in each group attended mathematics workshops dedicated to their age group:

Group A

• Martyna Górska, The mysterious world of prime numbers

  During our workshop we will talk about basic facts connected with prime numbers. We will try to understand their importance and the reason why prime numbers are still so interesting for mathematicians.

• Agnieszka Krause, Angles and circles

  We present some theorems connected with angles and circles, such as: Central and inscribed angles theorem, Cyclic quadrilateral theorem, Ptolemy’s theorem, Simson’s line. Moreover we say about Power of a Point theorem, Radical axis theorem and Radical center. Then, we use these theorems to solve some problems and observe some properties.

• Magdalena Wysokińska-Pliszka, On Pythagorean Theorem

  The goal of the lectures is to analyse some facts concerning the Pythagorean Theorem. First, in the form of a workshop, we will perform various geometrical proofs of it. Then we will focus on some interesting consequences and generalizations. Finally, our subject of interest will be the unusual applications of the Pythagorean Theorem in mathematical problems.

Group B

• Witold Kraśkiwicz, Convexity

  We take a closer look at convex sets in one, two and three dimensions. The workshop concentrates on Helly’s theorem and its applications.

• Mieczysław K. Mentzen, Conquest of infinity

  A sketch of the history of struggles with the concept of infinity will be presented. Next, the apparently surprising properties of infinite sets will be discussed.

• Mateusz Topolewski, Decision problems and mathematical games

  Life is full of decision problems. How to invest the money? Which car will be my best? Have breakfast or sleep 15 minutes more? All those problems can be regarded as mathematical games. In some games, the result is independent on our decisions. In some we need to be smart. Unfortunately, in most cases we need to decide having incomplete information and random reactions of other players. During the workshop we will try to learn how to make smart decisions.

Funny Mathematics

In addition to mathematical workshops focused on formulas, theorems, and proofs, participants had the opportunity to attend a different kind of session. Piotr Nodzyński demonstrated that mathematics also has an entertaining side, known to us as logic games, puzzles, and various types of brain teasers. During the workshop, he shared his extensive collection of games with the participants.

Mathematical Competitions

The camp participants, divided into groups A and B, took part in two mathematical competitions. Both were highly popular.

• Problem of the Day was a competition in which, for five consecutive days, participants solved a daily mathematical problem on their own between other planned activities. The solution had to be submitted the next morning.
• Main Competition was a traditional contest where participants, within a limited time, tackled three problems requiring the skills they had acquired during the workshops.

Closing Lecture

During the closing ceremony, Mateusz Topolewski delivered a short lecture titled How to Tame Randomness? (PDF), in which he contrasted the common understanding of randomness with the mathematical approach to the concept. Using simple examples, he also demonstrated that in the world of random phenomena, intuition can be misleading.

Tourism

A mathematician does not live by mathematics alone. Especially a young one. As part of the camp, participants also became tourists in Zakopane.

Mountain Hikes

The hike to Nosal took place as the first mountain walk of the camp. It was a good training session to test hiking skills. Kangaroos who did not manage to reach the mountain peak took a walk along Krupówki Street. They had the chance to try local specialties, for which the mountains are famous, including sheep’s milk cheese called oscypek. They could also take pictures with a white bear or meet a real highlander ;-)

Nosal is the northeastern end of Kasprowy Wierch. It belongs to the Tatra Mountains, where, according to legend, sleeping knights rest. Some managed to spot them. The rest surely admired the breathtaking mountain views.

Gubałówka. Besides the famous Krupówki Street, this is probably the best known place in Zakopane. After a short walk through the city to Ciągłówki, the Kangaroos followed the green trail uphill, past the old hospital, to the Kotelnica terrace, and further through Furmanowa to Gubałówka. From the viewpoint on the upper clearing, there is a stunning panorama of the Tatra Mountains. There is also a gravity slide, which the Kangaroos eagerly used. After a break, they headed west to the upper station of the chairlift on Butorowy Wierch. The descent on the chairlift took about 15 minutes, during which they could admire the view of the Western Tatras and Zakopane.

The last mountain hike, the most demanding but also the most rewarding - Hala Gąsienicowa and Czarny Staw Gąsienicowy. The camp participants took a bus to Kuźnice and from there hiked along the blue trail through Boczań to Hala Gąsienicowa and then further to Czarny Staw. On the way back, a mandatory stop at the Murowaniec shelter for a drink and a well-deserved treat for the tired hikers.

Zakopane Thermal Baths

After mountain adventures, to relax their muscles, the camp participants visited the Zakopane Thermal Baths. The hot springs greatly improve well-being and aid body regeneration.

Thermal baths were already used by the ancient Romans due to the healing properties of thermal waters. These are underground waters that, through the process of leaching rocks, contain a high concentration of mineral salts. The minerals in the water support skin regeneration, reduce headaches, and help relieve stress.

An additional attraction at the Zakopane Thermal Baths is the open-air rooftop pool, offering a beautiful panoramic view of the Tatra Mountains.

Kraków and Wieliczka

In addition to the attractions of Zakopane, camp participants took part in two full-day bus trips: to the Wieliczka Salt Mine and to Kraków.

Not much needs to be said about Kraków. The historic capital of Poland, featuring Wawel Hill and a picturesque old town, is rich in history and legends. The Kangaroos learned some of them during a guided walking tour. There was also time for photos with the Wawel Dragon and a stop for souvenirs and local delicacies.

The Wieliczka Salt Mine is one of the world’s greatest tourist attractions. Since 1978, it has been listed as a UNESCO World Cultural and Natural Heritage site. The mine has existed for over 700 years and was operational until June 30, 1996. Since then, it has been a purely tourist attraction, visited by nearly a million people every year. The tourist route includes 20 historic chambers connected by corridors with a total length of approximately 2 km. At the end of the route, at a depth of 135 meters, is the Kraków Saltworks Museum. The camp participants admired exhibits depicting the history of the mine, tools used for salt extraction and transportation, and the history of the city.

Sport and games

As the saying goes, mathematics is the queen of sciences. There is a lot of truth in this, as it appears in every aspect of our lives — from the construction of ancient pyramids to modern work on artificial intelligence. Mathematics also plays a role in entertainment, such as in board games.

Chess has been known since the 6th century AD, and since 2000, it has been officially recognized as a sport. There are many variations of the game, differing in the way matches are played. During the camp, participants took part in two tournaments: traditional individual chess and the Bughouse chess tournament. At the camp, chess serves as a mental exercise, enhances logical thinking, and helps integrate participants through a fair competition.

Table tennis tournaments also take place during the camp. Table tennis has been an officially recognized sport since 1988, and since 2004, it has been played in four categories: men's and women's singles, as well as men's and women's doubles. At the camp, singles tournaments for girls and boys were played using the Portuguese system — players were eliminated only after losing two matches. This year, for the first time, a tournament for camp leaders was also held.

Young Talents Contest

After returning from Kraków, a gala was held where participants showcased their, sometimes surprising and unexpected, talents. They demonstrated their skills in magic, dance, singing, and reenacting camp life.

More photos from the entire camp can be found in the gallery.