# Independent Groups in Federated Learning: Solution

*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. Answer and official solution.*

**Answer:** 6

**Solution.**

To create **as many** groups **as possible**, it pays to put *all* 10 channels *inside one* group and to leave the remaining devices one by one, separately. Then the channels will not “glue together” different groups.

How many devices are needed inside one group so that all 10 channels fit there? Inside a group of $t$ devices, the maximum number of channels is one between each pair, that is,

$$\text{maximum number of channels} = \frac{t \cdot (t - 1)}{2}.$$

$$t = 5: \ \frac{5 \cdot 4}{2} = 10.$$

Hence, we can make one large group of 5 devices and put all 10 channels there. The remaining $10 - 5 = 5$ devices stand one by one — that is 5 more groups.

Total number of groups: 1 large + 5 singletons = 6.

Why can there not be more than 6? If we try to make the large group out of fewer than 5 devices, at most 6 channels fit inside it; the remaining channels will inevitably start connecting different groups with each other, and the number of groups will decrease. Therefore, the maximum is $\boxed{6}$.