## Scatter Plot: Reasoning Tokens vs. Problem Size for o3-mini ### Overview The image is a scatter plot comparing the number of reasoning tokens used by the "o3-mini" model against the problem size. The plot distinguishes between successful and failed attempts, using blue circles for successful attempts and orange squares for failed attempts. The x-axis represents the problem size, and the y-axis represents the number of reasoning tokens. ### Components/Axes * **Title:** There is no explicit title on the chart. * **X-axis:** * Label: "Problem Size" * Scale: 0 to 400, with major ticks at 0, 100, 200, 300, and 400. * **Y-axis:** * Label: "Reasoning Tokens" * Scale: 0 to 50000, with major ticks at 0, 10000, 20000, 30000, 40000, and 50000. * **Legend:** Located in the top-right corner. * Blue circle: "o3-mini (Successful)" * Orange square: "o3-mini (Failed)" ### Detailed Analysis **o3-mini (Successful) - Blue Circles:** * **Trend:** The number of reasoning tokens generally increases with problem size for successful attempts. * **Data Points:** * Problem Size ~10, Reasoning Tokens ~2000 * Problem Size ~20, Reasoning Tokens ~3000 * Problem Size ~30, Reasoning Tokens ~4000 * Problem Size ~40, Reasoning Tokens ~6000 * Problem Size ~50, Reasoning Tokens ~8000 * Problem Size ~60, Reasoning Tokens ~9000 * Problem Size ~70, Reasoning Tokens ~12000 * Problem Size ~80, Reasoning Tokens ~13000 * Problem Size ~90, Reasoning Tokens ~23000 **o3-mini (Failed) - Orange Squares:** * **Trend:** For failed attempts, the number of reasoning tokens initially increases with problem size, but then appears to decrease or plateau as the problem size increases beyond approximately 100. * **Data Points:** * Problem Size ~20, Reasoning Tokens ~3000 * Problem Size ~40, Reasoning Tokens ~8000 * Problem Size ~60, Reasoning Tokens ~18000 * Problem Size ~80, Reasoning Tokens ~40000 * Problem Size ~100, Reasoning Tokens ~48000 * Problem Size ~120, Reasoning Tokens ~8000 * Problem Size ~140, Reasoning Tokens ~25000 * Problem Size ~160, Reasoning Tokens ~14000 * Problem Size ~180, Reasoning Tokens ~10000 * Problem Size ~200, Reasoning Tokens ~12000 * Problem Size ~220, Reasoning Tokens ~11000 * Problem Size ~260, Reasoning Tokens ~4000 * Problem Size ~280, Reasoning Tokens ~6000 * Problem Size ~300, Reasoning Tokens ~11000 * Problem Size ~380, Reasoning Tokens ~8000 * Problem Size ~400, Reasoning Tokens ~12000 ### Key Observations * For successful attempts, there is a clear positive correlation between problem size and the number of reasoning tokens. * For failed attempts, the relationship is more complex. Initially, the number of reasoning tokens increases with problem size, but beyond a certain point (around 100), the number of tokens used in failed attempts appears to decrease or plateau. * There is a significant difference in the number of reasoning tokens used between successful and failed attempts for smaller problem sizes. ### Interpretation The data suggests that for the "o3-mini" model, successful problem-solving generally requires more reasoning tokens as the problem size increases. However, when the model fails, it may be due to an inefficient use of reasoning tokens, especially for larger problem sizes. The plateau or decrease in reasoning tokens for failed attempts at larger problem sizes could indicate that the model is either giving up early or getting stuck in a loop, failing to explore the solution space effectively. The model may be more likely to fail when the problem size is larger, and the number of reasoning tokens is lower. This could be due to the model not having enough resources to solve the problem.

\n ## Scatter Plot: Reasoning Tokens vs. Problem Size ### Overview This image presents a scatter plot visualizing the relationship between "Problem Size" and "Reasoning Tokens" for two categories: successful and failed attempts, both labeled as "o3-mini". The plot aims to show how the number of reasoning tokens used changes with the size of the problem, and whether success or failure correlates with token usage. ### Components/Axes * **X-axis:** "Problem Size" - ranging from approximately 0 to 420. * **Y-axis:** "Reasoning Tokens" - ranging from 0 to 52000. * **Legend:** Located in the top-right corner. * Blue circles: "o3-mini (Successful)" * Orange squares: "o3-mini (Failed)" * **Grid:** A light gray grid is present, aiding in the reading of values. ### Detailed Analysis The plot contains two distinct data series: successful and failed "o3-mini" attempts. **Successful Attempts (Blue Circles):** The trend for successful attempts is generally upward, but with significant variation. * At a Problem Size of approximately 10, Reasoning Tokens are around 2000. * As Problem Size increases to around 80, Reasoning Tokens increase to approximately 22000. * Around a Problem Size of 100, Reasoning Tokens reach a peak of around 25000. * From Problem Size 100 to 400, Reasoning Tokens fluctuate between 5000 and 15000, with a general decreasing trend. **Failed Attempts (Orange Squares):** The trend for failed attempts is more scattered and generally shows a decrease in Reasoning Tokens as Problem Size increases. * At a Problem Size of approximately 10, Reasoning Tokens are around 1000. * Between Problem Sizes of 50 and 150, Reasoning Tokens vary widely, ranging from approximately 8000 to 50000. * From Problem Size 200 to 400, Reasoning Tokens generally decrease, fluctuating between 5000 and 10000. * There is a notable outlier at a Problem Size of approximately 120, with Reasoning Tokens around 52000. ### Key Observations * **Positive Correlation (Successful):** There's a positive correlation between Problem Size and Reasoning Tokens for successful attempts, up to a certain point (around Problem Size 100). Beyond that, the correlation weakens. * **Negative Correlation (Failed):** There's a general negative correlation between Problem Size and Reasoning Tokens for failed attempts, though it's less consistent. * **Outlier:** The failed attempt at a Problem Size of approximately 120 with 52000 Reasoning Tokens is a significant outlier. * **Token Usage:** Successful attempts generally use fewer tokens than failed attempts for larger problem sizes. ### Interpretation The data suggests that for smaller problem sizes, successful "o3-mini" attempts require more reasoning tokens. This could indicate that the algorithm needs to explore more possibilities to find a solution when the problem is relatively simple. However, as the problem size increases, the number of tokens needed for success decreases, potentially because the problem becomes more constrained or the algorithm converges more quickly. The failed attempts show a more erratic pattern, with a high outlier suggesting a case where the algorithm spent a significant amount of resources without finding a solution. The general decrease in token usage for failed attempts with increasing problem size could indicate that the algorithm gives up more quickly on larger problems, or that the search space becomes less navigable. The difference in token usage between successful and failed attempts, particularly for larger problem sizes, suggests that there's a threshold of reasoning effort beyond which the algorithm is unlikely to succeed. The data could be used to optimize the algorithm's resource allocation, potentially by setting a maximum token limit or by dynamically adjusting the search strategy based on problem size.

## Scatter Plot: Reasoning Tokens vs. Problem Size for o3-mini ### Overview This is a scatter plot comparing the number of "Reasoning Tokens" used against "Problem Size" for a model or system identified as "o3-mini." The data is split into two categories: successful runs and failed runs. The plot reveals a distinct separation in the distribution and behavior of these two categories. ### Components/Axes * **Chart Type:** Scatter Plot * **X-Axis:** * **Label:** "Problem Size" * **Scale:** Linear, ranging from 0 to 400. * **Major Tick Marks:** 0, 100, 200, 300, 400. * **Y-Axis:** * **Label:** "Reasoning Tokens" * **Scale:** Linear, ranging from 0 to 50,000. * **Major Tick Marks:** 0, 10000, 20000, 30000, 40000, 50000. * **Legend:** * **Position:** Top-right corner of the plot area. * **Series 1:** Blue circle marker, labeled "o3-mini (Successful)". * **Series 2:** Orange square marker, labeled "o3-mini (Failed)". * **Grid:** Light gray gridlines are present for both axes. ### Detailed Analysis **1. o3-mini (Successful) - Blue Circles:** * **Spatial Grounding & Trend:** These data points are tightly clustered in the bottom-left quadrant of the plot. The trend shows a positive correlation: as Problem Size increases from approximately 10 to 80, the Reasoning Tokens used also increase, but remain below ~25,000. * **Data Point Distribution (Approximate):** * Problem Size ~10-30: Tokens range from ~2,000 to ~8,000. * Problem Size ~30-60: Tokens range from ~8,000 to ~15,000. * Problem Size ~60-80: Tokens range from ~10,000 to ~24,000. The highest token count for a successful run is approximately 24,000 at a Problem Size of about 75. * **Key Characteristic:** No successful runs are plotted for Problem Sizes greater than approximately 80. **2. o3-mini (Failed) - Orange Squares:** * **Spatial Grounding & Trend:** These points are widely dispersed across the entire plot area. There is no single clear linear trend. The distribution suggests failures can occur with both low and high token usage across a wide range of problem sizes. * **Data Point Distribution (Approximate):** * **High-Token Failures (Outliers):** Several failures occur at relatively small Problem Sizes (50-150) but with extremely high token counts, including the highest points on the chart: ~54,000 tokens at Problem Size ~100 and ~48,000 tokens at Problem Size ~90. * **Mid-Range Failures:** A cluster exists between Problem Size 100-250 with token counts scattered between ~5,000 and ~30,000. * **Large-Problem Failures:** Failures are recorded for Problem Sizes up to 400. At these larger sizes (300-400), the token counts are generally lower, mostly between ~3,000 and ~12,000. * **Key Characteristic:** Failed runs exist across the entire spectrum of Problem Size (from ~50 to 400) and Reasoning Tokens (from ~3,000 to ~54,000). ### Key Observations 1. **Clear Separation by Problem Size:** Successful runs are confined to Problem Sizes below ~80. All runs with Problem Size >80 are failures. 2. **Inverse Relationship for Failures at Extremes:** The highest token usage (potential overthinking/inefficiency) occurs for failures on moderately sized problems (~50-150). Failures on the largest problems (~300-400) use comparatively fewer tokens. 3. **Absence of Successful Large Problems:** The chart shows no data points for successful runs on problems larger than ~80, indicating a potential capability boundary for the o3-mini model in this test. 4. **Token Usage Variability:** Failed runs exhibit vastly greater variability in token consumption compared to the more predictable, lower-token successful runs. ### Interpretation The data suggests a strong correlation between problem complexity (size) and the model's ability to succeed, with a clear threshold around a Problem Size of 80. The "Successful" series demonstrates efficient scaling: token usage grows moderately with problem size. The "Failed" series tells a more complex story. The cluster of high-token failures at moderate problem sizes may indicate scenarios where the model engaged in extensive but unproductive reasoning, ultimately failing. Conversely, failures at very large problem sizes with lower token counts might suggest the model gave up early or failed to initiate a sufficiently deep reasoning process. **Peircean Investigation:** The chart acts as a sign of the model's **performance envelope**. The tight blue cluster is an *icon* of efficient, successful processing within its comfort zone. The scattered orange squares are an *index* of failure modes: some point to inefficiency (high tokens), others to insufficient effort (low tokens on large problems). The stark boundary at Problem Size ~80 is a *symbol* of a hard limit in the model's tested capability. This visualization is crucial for diagnosing whether failures stem from computational inefficiency or fundamental inability.

## Scatter Plot: Reasoning Tokens vs. Problem Size for o3-mini ### Overview The image is a scatter plot comparing the number of reasoning tokens used by the o3-mini model against problem size, differentiated by success/failure outcomes. Two data series are represented: blue circles for successful runs and orange squares for failed runs. The plot spans problem sizes from 0 to 400 and reasoning tokens from 0 to 50,000. ### Components/Axes - **X-axis (Problem Size)**: Labeled "Problem Size," ranging from 0 to 400 in increments of 100. - **Y-axis (Reasoning Tokens)**: Labeled "Reasoning Tokens," ranging from 0 to 50,000 in increments of 10,000. - **Legend**: Located in the top-right corner, with: - **Blue circles**: "o3-mini (Successful)" - **Orange squares**: "o3-mini (Failed)" ### Detailed Analysis - **Blue Circles (Successful)**: - Clustered primarily in the lower-left quadrant. - Problem sizes range from ~0 to ~100. - Reasoning tokens range from ~0 to ~25,000. - Density decreases as problem size increases. - **Orange Squares (Failed)**: - Distributed across the entire plot but concentrated in the upper-right quadrant. - Problem sizes range from ~50 to ~400. - Reasoning tokens range from ~5,000 to ~50,000. - Notable outliers: A few orange squares at problem size ~100 with reasoning tokens ~50,000. ### Key Observations 1. **Successful Runs**: Dominated by smaller problem sizes (<100) and lower token usage (<25,000). 2. **Failed Runs**: More variable, with larger problem sizes (>200) and higher token usage (>10,000). 3. **Outliers**: A small subset of failed runs at problem size ~100 required ~50,000 tokens, suggesting inefficiency or edge cases. ### Interpretation The data suggests that successful reasoning by o3-mini is strongly correlated with smaller problem sizes and efficient token usage. Failed runs, however, exhibit a broader range of problem sizes and token consumption, indicating potential challenges with scalability or resource allocation. The outliers at problem size ~100 with high token usage may represent edge cases where the model struggles despite smaller inputs, possibly due to algorithmic complexity or data quality issues. This highlights a trade-off between problem size, computational resources, and success rates for the o3-mini model.