## Bar Chart: Number of Theorem Attempts It Took to Solve a Theorem
### Overview
This is a grouped bar chart comparing the performance of two models, "Sonnet" and "Gemini," on a theorem-solving task. The chart displays the frequency distribution of the number of attempts required by each model to successfully solve a theorem. The data suggests a right-skewed distribution for both models, with the vast majority of theorems being solved in a very low number of attempts.
### Components/Axes
* **Chart Title:** "Number of theorem attempts it took to solve a theorem"
* **X-Axis:** Labeled "Number of Theorem Attempts". It is a linear scale with major tick marks at 0, 5, 10, 15, 20, and 25. The axis represents discrete counts of attempts.
* **Y-Axis:** Labeled "Number of Samples". It is a linear scale with major tick marks at 0, 10, 20, 30, 40, and 50. The axis represents the frequency or count of theorems solved within a specific number of attempts.
* **Legend:** Located in the top-right corner of the plot area.
* A blue square corresponds to the label "Sonnet".
* An orange square corresponds to the label "Gemini".
* **Data Series:** Two sets of vertical bars, grouped by the x-axis value (number of attempts). For each x-value, the blue bar (Sonnet) is on the left, and the orange bar (Gemini) is on the right.
### Detailed Analysis
The following table reconstructs the approximate data points from the chart. Values are estimated based on bar height relative to the y-axis.
| Number of Attempts (X) | Sonnet (Blue) - Approx. Sample Count | Gemini (Orange) - Approx. Sample Count |
| :--- | :--- | :--- |
| 0 | ~2 | ~6 |
| 1 | **~52** | **~38** |
| 2 | ~8 | ~4 |
| 3 | ~12 | ~2 |
| 4 | ~9 | ~3 |
| 5 | ~2 | ~1 |
| 6 | ~4 | ~1 |
| 7 | ~2 | ~2 |
| 8 | ~3 | ~4 |
| 9 | ~4 | ~2 |
| 10 | ~3 | ~1 |
| 11 | ~0 | ~1 |
| 12 | ~0 | ~1 |
| 13 | ~2 | ~1 |
| 14 | ~1 | ~2 |
| 15 | ~4 | ~1 |
| 16 | ~1 | ~3 |
| 17 | ~0 | ~1 |
| 18 | ~3 | ~1 |
| 19 | ~1 | ~3 |
| 20 | ~0 | ~0 |
| 21 | ~0 | ~0 |
| 22 | ~1 | ~1 |
| 23 | ~1 | ~4 |
| 24 | ~2 | ~0 |
| 25 | ~0 | ~0 |
**Trend Verification:**
* **Sonnet (Blue):** The distribution has a very sharp peak at 1 attempt, followed by a rapid decline. There is a secondary, much smaller cluster of activity between 2-4 attempts. The tail extends thinly and sporadically out to 24 attempts.
* **Gemini (Orange):** Also peaks at 1 attempt, but the peak is lower than Sonnet's. Its decline after the peak is more gradual, with a more noticeable presence in the 2-9 attempt range compared to Sonnet. Its tail is similarly sparse but includes a small, notable uptick at 23 attempts.
### Key Observations
1. **Dominant Peak at 1 Attempt:** For both models, the single most common outcome is solving the theorem on the first attempt. Sonnet has a significantly higher count (~52) at this point compared to Gemini (~38).
2. **Rapid Drop-off:** The number of samples drops dramatically for both models after 1 attempt. By 5 attempts, the counts are in the low single digits.
3. **Long Tail:** Both distributions have a long, sparse tail extending to over 20 attempts, indicating that a small subset of theorems are significantly more difficult and require many attempts.
4. **Comparative Performance:** Sonnet appears to have a higher concentration of successes at the very lowest attempt counts (0-4), while Gemini shows a slightly more distributed performance across the 2-9 attempt range.
5. **Notable Outlier:** Gemini shows a small but distinct increase in samples at 23 attempts (~4), which is higher than its counts for the preceding ~10 attempt values.
### Interpretation
This chart visualizes the efficiency and difficulty distribution of a theorem-solving task for two AI models. The data suggests that the task is not uniformly difficult; most theorems are "easy" and solvable in one or a few attempts, while a minority are "hard," requiring an order of magnitude more effort.
The pronounced peak at 1 attempt for both models indicates a high baseline proficiency on a significant portion of the test set. The difference in peak height (Sonnet > Gemini) could suggest Sonnet is more effective on the easiest problems or that the test set contained more problems aligned with Sonnet's strengths.
The long tail is critical from a research perspective. It highlights the models' limitations and the existence of challenging edge cases. The small cluster of Gemini results at 23 attempts might indicate a specific class of problem where Gemini's strategy leads to prolonged, but eventually successful, exploration, whereas Sonnet either solves it quicker or fails (if failures are not plotted). The chart does not show failed attempts, only successful ones, so we cannot determine the total failure rate.
In summary, the chart demonstrates that while both models are highly effective on the majority of theorems, their performance diverges on moderately difficult problems, and both encounter a subset of problems that are exceptionally challenging, requiring over 20 attempts to solve.
