# Independent Groups in Federated Learning

*English translation by SOTA – AI Community of the Russian original. Organisers who would like this translation removed can email sota.ai.community@gmail.com.*

*Task 1 of the school stage of the All-Russian School Olympiad (VsOSh) 2025/26 in artificial intelligence (region group III), grades 9–11. The official answer and solution are in a separate file.*

In collaborative learning (it is also called federated learning), many devices learn together: each one computes locally and sends only a small share of the total volume of numerical data. For this exchange, two-way communication channels have been laid between some pairs of devices. The task is performed by 10 devices. Two-way channels have been laid between some pairs of devices, 10 channels in total (there is never more than one channel between the same pair of devices). We call a group of devices an independent group if any device in it can be reached from any other via a chain of channels, and there is not a single channel between different such groups. What is the largest number of independent groups that can be obtained under these conditions?

**Scoring criterion:** exact match of the answer — 12 points

**Maximum score for the task — 12**