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Comparing the answers between Openai and Gemini the writing style of Gemini is a lot clearer. It could be presented a bit better but it's easy enough to follow

Comparing the answers between Openai and Gemini the writing style of Gemini is a lot clearer. It could be presented a bit better but it's easy enough to follow the proof. This also makes it a lot shorter than the answer given by OpenAI and it uses proper prose.

cubefox 1y ago
I found the proofs you were referring to: Google https://storage.googleapis.com/deepmind-media/gemini/IMO_2025.pdf https://storage.googleapis.com/deepmind-media/gemini/IMO_202... OpenAI https://github.com/aw31/openai-imo-2025-proofs/ https://github.com/aw31/openai-imo-2025-proofs/
- sweezyjeezy 1y ago
  Gemini is clearer but MY GOD is it verbose. e.g. look at problem 1, section 2. Analysis of the Core Problem - there's nothing at all deep here, but it seems the model wants to spell out every single tiny logical step. I wonder if this is a stylistic choice or something that actually helps the model get to the end.
  - shiandow 1y ago
    Section 2 is a case by case analysis. Those are never pretty but perfectly normal given the problem. With OpenAI that part takes up about 2/3 if the proof even with its fragmented prose. I don't think it does much better.
    - sweezyjeezy 1y ago
      It's not it being case by case that's my issue. I used do olympiads and e.g. for the k>=3 case I wouldn't write much more than: "Since there are 3k - 3 points on the perimeter of the triangle to be covered, and any sunny line can pass through at most two of them, it follows that 3k − 3 ≤ 2k, i.e. k ≤ 3." Gemini writes: Let Tk be the convex hull of Pk. Tk is the triangle with vertices V1 = (1, 1), V2 = (1, k), V3 = (k, 1). The edges of Tk lie on the lines x = 1 (V), y = 1 (H), and x + y = k + 1 (D). These lines are shady. Let Bk be the set of points in Pk lying on the boundary of Tk. Each edge contains k points. Since the vertices are distinct (as k ≥ 2), the total number of points on the boundary is |Bk| = 3k − 3. Suppose Pk is covered by k sunny lines Lk. These lines must cover Bk. Let L ∈ Lk. Since L is sunny, it does not coincide with the lines containing the edges of Tk. A line that does not contain an edge of a convex polygon intersects the boundary of the polygon at most at two points. Thus, |L ∩ Bk| ≤ 2. The total coverage of Bk by Lk is at most 2k. We must have |Bk| ≤ 2k. 3k − 3 ≤ 2k, which implies k ≤ 3.
      - shiandow 1y ago
        I'll admit I didn't look to deeply if it could be done simpler, but surely that is still miles better than what OpenAI did? At least Gemini's can be simplified. OpenAI labels all points and then enumerates all the lines that go through them.
  - vessenes 1y ago
    They actually do help - in that they give the model more computation time and also allow realtime management of the input context by the model. You can see this same behavior in the excessive comment writing some coding models engage in; Anthropic interviews said these do actually help the model.
    - johnfn 1y ago
      Gemini did not one-shot these answers; it did its thinking elsewhere (probably not released by Google) and then it consolidated it down into what you see in the PDF. From the article: > We achieved this year’s result using an advanced version of Gemini Deep Think – an enhanced reasoning mode for complex problems that incorporates some of our latest research techniques, including parallel thinking. This setup enables the model to simultaneously explore and combine multiple possible solutions before giving a final answer, rather than pursuing a single, linear chain of thought. I don't see any parallel thinking, e.g., so that was probably elided in the final results.
      - noahgav 1y ago
        Yes, because these are the answers it gave, not the thinking.
- CamperBob2 1y ago
  Kind of disappointing that neither provider shows the unsuccessful attack on problem 6.
  - cubefox 1y ago
    They don't show any reasoning traces at all, just the final proofs. We must assume the traces are pretty huge, since at least Google makes it clear that they are heavily relying on inference compute: > We achieved this year’s result using an advanced version of Gemini Deep Think – an enhanced reasoning mode for complex problems that incorporates some of our latest research techniques, including parallel thinking. This setup enables the model to simultaneously explore and combine multiple possible solutions before giving a final answer, rather than pursuing a single, linear chain of thought. [...] We will be making a version of this Deep Think model available to a set of trusted testers, including mathematicians, before rolling it out to Google AI Ultra subscribers.
aabhay 1y ago
Based on the Google presented answers, its possible that the report is generated post-hoc as a summarization of the prior thoughts. One could also presume that this summarization step is part of the mechanism for running the Tree of Thoughts too, so that this wasn’t some manual “now give the final answer” command.

cubefox 1y ago

I found the proofs you were referring to: Google https://storage.googleapis.com/deepmind-media/gemini/IMO_2025.pdf https://storage.googleapis.com/deepmind-media/gemini/IMO_202... OpenAI https://github.com/aw31/openai-imo-2025-proofs/ https://github.com/aw31/openai-imo-2025-proofs/

sweezyjeezy 1y ago
Gemini is clearer but MY GOD is it verbose. e.g. look at problem 1, section 2. Analysis of the Core Problem - there's nothing at all deep here, but it seems the model wants to spell out every single tiny logical step. I wonder if this is a stylistic choice or something that actually helps the model get to the end.
- shiandow 1y ago
  Section 2 is a case by case analysis. Those are never pretty but perfectly normal given the problem. With OpenAI that part takes up about 2/3 if the proof even with its fragmented prose. I don't think it does much better.
  - sweezyjeezy 1y ago
    It's not it being case by case that's my issue. I used do olympiads and e.g. for the k>=3 case I wouldn't write much more than: "Since there are 3k - 3 points on the perimeter of the triangle to be covered, and any sunny line can pass through at most two of them, it follows that 3k − 3 ≤ 2k, i.e. k ≤ 3." Gemini writes: Let Tk be the convex hull of Pk. Tk is the triangle with vertices V1 = (1, 1), V2 = (1, k), V3 = (k, 1). The edges of Tk lie on the lines x = 1 (V), y = 1 (H), and x + y = k + 1 (D). These lines are shady. Let Bk be the set of points in Pk lying on the boundary of Tk. Each edge contains k points. Since the vertices are distinct (as k ≥ 2), the total number of points on the boundary is |Bk| = 3k − 3. Suppose Pk is covered by k sunny lines Lk. These lines must cover Bk. Let L ∈ Lk. Since L is sunny, it does not coincide with the lines containing the edges of Tk. A line that does not contain an edge of a convex polygon intersects the boundary of the polygon at most at two points. Thus, |L ∩ Bk| ≤ 2. The total coverage of Bk by Lk is at most 2k. We must have |Bk| ≤ 2k. 3k − 3 ≤ 2k, which implies k ≤ 3.
    - shiandow 1y ago
      I'll admit I didn't look to deeply if it could be done simpler, but surely that is still miles better than what OpenAI did? At least Gemini's can be simplified. OpenAI labels all points and then enumerates all the lines that go through them.
- vessenes 1y ago
  They actually do help - in that they give the model more computation time and also allow realtime management of the input context by the model. You can see this same behavior in the excessive comment writing some coding models engage in; Anthropic interviews said these do actually help the model.
  - johnfn 1y ago
    Gemini did not one-shot these answers; it did its thinking elsewhere (probably not released by Google) and then it consolidated it down into what you see in the PDF. From the article: > We achieved this year’s result using an advanced version of Gemini Deep Think – an enhanced reasoning mode for complex problems that incorporates some of our latest research techniques, including parallel thinking. This setup enables the model to simultaneously explore and combine multiple possible solutions before giving a final answer, rather than pursuing a single, linear chain of thought. I don't see any parallel thinking, e.g., so that was probably elided in the final results.
    - noahgav 1y ago
      Yes, because these are the answers it gave, not the thinking.
CamperBob2 1y ago
Kind of disappointing that neither provider shows the unsuccessful attack on problem 6.
- cubefox 1y ago
  They don't show any reasoning traces at all, just the final proofs. We must assume the traces are pretty huge, since at least Google makes it clear that they are heavily relying on inference compute: > We achieved this year’s result using an advanced version of Gemini Deep Think – an enhanced reasoning mode for complex problems that incorporates some of our latest research techniques, including parallel thinking. This setup enables the model to simultaneously explore and combine multiple possible solutions before giving a final answer, rather than pursuing a single, linear chain of thought. [...] We will be making a version of this Deep Think model available to a set of trusted testers, including mathematicians, before rolling it out to Google AI Ultra subscribers.

aabhay 1y ago

Based on the Google presented answers, its possible that the report is generated post-hoc as a summarization of the prior thoughts. One could also presume that this summarization step is part of the mechanism for running the Tree of Thoughts too, so that this wasn’t some manual “now give the final answer” command.