# Falsifying a Hypothesis: 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 4 of the school stage of the All-Russian School Olympiad (VsOSh) 2025/26 in artificial intelligence (region group IV), grades 9–11. Answer and official solution.*

**Answer:** (-3,-3,-3) and (-3,-2,-1)

**Solution.**

| Rule | $(1, 2, 4)$ | $(1, 2, 3)$ | $(-3, -3, -3)$ | $(-3, -2, -1)$ |
|:---:|:---:|:---:|:---:|:---:|
| 1 strictly increasing | Yes | Yes | No | Yes |
| 2 all positive | Yes | Yes | No | No |
| 3 any three | Yes | Yes | Yes | Yes |
| 4 $ac = b^2$ | Yes | No | Yes | No |

The first triples do not work:

- For (1,2,4) all the answers are “Yes”, so after the first question all 4 rules remain. With the one remaining question one can only get “Yes” or “No”, that is, split the rules into at most 2 groups, not distinguish all 4.
- For (1,2,3), 3 rules may remain after the first question (three “Yes” answers). With one remaining question, three options cannot be distinguished either.

The triples $(-3, -3, -3)$ and $(-3, -2, -1)$ work: the pairs of answers for the four rules are all different — (No, Yes), (No, No), (Yes, Yes), (Yes, No) — and they determine the rule unambiguously.