## Scatter Plot: Reasoning Tokens vs. Problem Size ### Overview The image is a scatter plot showing the relationship between "Problem Size" and "Reasoning Tokens" for three different levels of "Reasoning Effort": low, medium, and high. Each level has a scatter plot of data points and a corresponding fitted line with an R-squared value indicating the goodness of fit. ### Components/Axes * **X-axis:** "Problem Size", ranging from 0 to 100, with tick marks at intervals of 20. * **Y-axis:** "Reasoning Tokens", ranging from 0 to 50000, with tick marks at intervals of 10000. * **Legend (top-left):** * "Reasoning Effort" * Blue circle: "low" * Blue solid line: "low fit (R^2: 0.489)" * Orange square: "medium" * Orange dashed line: "medium fit (R^2: 0.833)" * Green triangle: "high" * Green dash-dot line: "high fit (R^2: 0.813)" ### Detailed Analysis * **Low Reasoning Effort (Blue):** * Data points: Scatter points are clustered near the bottom of the chart. * Trend: The "low fit" line is nearly flat, showing a slight positive slope. * Data points: (20, 1000), (30, 2000), (40, 2500) * R-squared: 0.489 * **Medium Reasoning Effort (Orange):** * Data points: Scatter points are in the middle range of the chart. * Trend: The "medium fit" line has a moderate positive slope. * Data points: (20, 1000), (30, 8000), (40, 9000), (70, 22000), (80, 14000) * R-squared: 0.833 * **High Reasoning Effort (Green):** * Data points: Scatter points are spread across the entire range of the chart. * Trend: The "high fit" line has a steep positive slope. * Data points: (20, 6000), (30, 10000), (40, 12000), (50, 35000), (60, 23000), (70, 28000), (80, 50000), (90, 40000), (100, 45000) * R-squared: 0.813 ### Key Observations * As "Problem Size" increases, "Reasoning Tokens" generally increase for all levels of "Reasoning Effort". * The "high" reasoning effort exhibits the most significant increase in "Reasoning Tokens" as "Problem Size" increases. * The R-squared values indicate that the fitted lines for "medium" and "high" reasoning effort are a better fit for the data than the "low" reasoning effort. ### Interpretation The plot demonstrates that the amount of reasoning required (measured in tokens) increases with problem size. The level of reasoning effort significantly impacts the rate at which reasoning tokens increase. Problems requiring "high" reasoning effort show a much steeper increase in tokens compared to "medium" or "low" effort problems. The R-squared values suggest that a linear model is more appropriate for "medium" and "high" reasoning effort than for "low" reasoning effort, where the relationship between problem size and reasoning tokens may be more complex or less pronounced. The "low" reasoning effort shows a very weak correlation, suggesting that problem size has little impact on the number of reasoning tokens required when the reasoning effort is low.

\n ## Scatter Plot: Reasoning Tokens vs. Problem Size ### Overview This image presents a scatter plot illustrating the relationship between "Problem Size" (on the x-axis) and "Reasoning Tokens" (on the y-axis) for three different levels of "Reasoning Effort" (low, medium, and high). Each effort level is represented by a different color and a corresponding trendline. The plot aims to demonstrate how the amount of reasoning required (measured in tokens) scales with the complexity of the problem, categorized by the level of effort involved. ### Components/Axes * **X-axis:** "Problem Size" - Scale ranges from approximately 15 to 105, with markings at 20, 40, 60, 80, and 100. * **Y-axis:** "Reasoning Tokens" - Scale ranges from 0 to 55,000, with markings at 0, 10,000, 20,000, 30,000, 40,000, and 50,000. * **Legend:** Located in the top-right corner. * "low" - Represented by blue circles. * "medium" - Represented by orange squares. * "high" - Represented by green triangles. * **Trendlines:** * "low fit (R^2: 0.489)" - Solid blue line. * "medium fit (R^2: 0.833)" - Dashed orange line. * "high fit (R^2: 0.813)" - Dashed green line. * **Data Points:** Scatter points representing individual data instances for each reasoning effort level. ### Detailed Analysis **Low Reasoning Effort (Blue):** The blue data points are scattered relatively close to the x-axis, with most values between 0 and 5,000 tokens. The trendline is nearly flat, indicating a minimal increase in reasoning tokens with increasing problem size. * At Problem Size = 20, Reasoning Tokens ≈ 1,000 * At Problem Size = 40, Reasoning Tokens ≈ 2,000 * At Problem Size = 60, Reasoning Tokens ≈ 3,000 * At Problem Size = 80, Reasoning Tokens ≈ 3,000 * At Problem Size = 100, Reasoning Tokens ≈ 4,000 **Medium Reasoning Effort (Orange):** The orange data points show a more pronounced upward trend than the low effort level. The trendline is steeper, indicating a more significant increase in reasoning tokens with increasing problem size. * At Problem Size = 20, Reasoning Tokens ≈ 4,000 * At Problem Size = 40, Reasoning Tokens ≈ 7,000 * At Problem Size = 60, Reasoning Tokens ≈ 12,000 * At Problem Size = 80, Reasoning Tokens ≈ 22,000 * At Problem Size = 100, Reasoning Tokens ≈ 28,000 **High Reasoning Effort (Green):** The green data points exhibit the strongest upward trend, with values ranging from approximately 5,000 to 50,000 tokens. The trendline is the steepest, indicating a substantial increase in reasoning tokens with increasing problem size. * At Problem Size = 20, Reasoning Tokens ≈ 7,000 * At Problem Size = 40, Reasoning Tokens ≈ 15,000 * At Problem Size = 60, Reasoning Tokens ≈ 25,000 * At Problem Size = 80, Reasoning Tokens ≈ 40,000 * At Problem Size = 100, Reasoning Tokens ≈ 52,000 The R^2 values indicate the goodness of fit for each trendline: 0.489 for low, 0.833 for medium, and 0.813 for high. Higher R^2 values suggest a stronger linear relationship between problem size and reasoning tokens. ### Key Observations * The relationship between problem size and reasoning tokens is strongly influenced by the reasoning effort level. * The "low" effort level shows the weakest correlation (lowest R^2 value) and minimal increase in reasoning tokens with problem size. * The "medium" and "high" effort levels exhibit strong correlations (high R^2 values) and significant increases in reasoning tokens with problem size. * The "high" effort level consistently requires the most reasoning tokens for any given problem size. * There is some scatter in the data points around the trendlines, indicating variability in the reasoning token requirements for individual problems within each effort level. ### Interpretation The data suggests that the amount of reasoning required to solve a problem increases with both the problem's size and the level of reasoning effort applied. The R^2 values indicate that the relationship is more linear and predictable for medium and high reasoning effort levels. The low effort level shows a weaker relationship, possibly because simpler strategies are employed that are less sensitive to problem size. The increasing trendlines for medium and high effort levels suggest that as problems become more complex, more sophisticated reasoning processes are needed, leading to a greater demand for reasoning tokens. This could be due to the need for more steps, more complex calculations, or more extensive search through possible solutions. The scatter around the trendlines indicates that there is inherent variability in the reasoning process. Even for problems of the same size and effort level, the exact number of reasoning tokens required can vary depending on the specific problem instance and the approach taken. This variability highlights the complexity of the reasoning process and the challenges in accurately predicting its resource requirements. The data could be used to estimate the computational resources needed for solving problems of different sizes and complexities, depending on the desired level of reasoning effort.

## Scatter Plot with Linear Regression Fits: Reasoning Tokens vs. Problem Size by Reasoning Effort ### Overview This image is a scatter plot chart displaying the relationship between "Problem Size" (x-axis) and the number of "Reasoning Tokens" (y-axis) used. The data is categorized into three levels of "Reasoning Effort": low, medium, and high. Each category includes individual data points and a fitted linear regression line with its corresponding R-squared value. ### Components/Axes * **Chart Title:** None visible. * **X-Axis:** * **Label:** "Problem Size" * **Scale:** Linear, ranging from approximately 15 to 100. Major tick marks are at 20, 40, 60, 80, and 100. * **Y-Axis:** * **Label:** "Reasoning Tokens" * **Scale:** Linear, ranging from 0 to over 50,000. Major tick marks are at 0, 10000, 20000, 30000, 40000, and 50000. * **Legend:** Located in the top-left quadrant of the plot area. It defines three data series and their corresponding fit lines: 1. **low:** Blue circle marker. 2. **low fit (R^2: 0.489):** Solid blue line. 3. **medium:** Orange square marker. 4. **medium fit (R^2: 0.833):** Dashed orange line. 5. **high:** Green triangle marker. 6. **high fit (R^2: 0.813):** Dash-dot green line. * **Grid:** A light gray grid is present in the background. ### Detailed Analysis The analysis is segmented by the three "Reasoning Effort" categories. **1. Low Reasoning Effort (Blue Circles & Solid Blue Line)** * **Trend:** The data points show a very shallow, slightly positive slope. The fitted line increases minimally from left to right. * **Data Points (Approximate):** * At Problem Size ~18: ~1,000 tokens. * At Problem Size ~20: ~1,500 tokens. * At Problem Size ~25: ~2,000 tokens. * At Problem Size ~30: ~2,500 tokens. * At Problem Size ~35: ~3,000 tokens. * At Problem Size ~42: ~3,500 tokens. * **Fit Line:** The solid blue regression line starts near (18, 1000) and ends near (42, 3500). The R-squared value of 0.489 indicates a weak to moderate fit, suggesting the linear model explains less than half of the variance in the data for this category. **2. Medium Reasoning Effort (Orange Squares & Dashed Orange Line)** * **Trend:** The data points and the fitted line show a clear, moderate positive linear trend. The slope is steeper than the "low" effort series. * **Data Points (Approximate):** * At Problem Size ~18: ~2,500 tokens. * At Problem Size ~20: ~3,000 tokens. * At Problem Size ~25: ~4,000 tokens. * At Problem Size ~30: ~5,500 tokens. * At Problem Size ~35: ~7,000 tokens. * At Problem Size ~42: ~8,500 tokens. * At Problem Size ~50: ~9,000 tokens. * At Problem Size ~55: ~12,000 tokens. * At Problem Size ~64: ~10,000 tokens. * At Problem Size ~72: ~23,000 tokens (potential outlier, high). * At Problem Size ~80: ~13,000 tokens and ~19,000 tokens. * **Fit Line:** The dashed orange regression line starts near (18, 2500) and ends near (80, 18000). The R-squared value of 0.833 indicates a strong fit, meaning the linear model explains a large portion of the variance. **3. High Reasoning Effort (Green Triangles & Dash-Dot Green Line)** * **Trend:** The data points and the fitted line show a strong, steep positive linear trend. This series has the steepest slope and the highest token counts. * **Data Points (Approximate):** The data is more scattered, especially at higher problem sizes. * At Problem Size ~18: ~5,000 tokens. * At Problem Size ~20: ~7,000 tokens. * At Problem Size ~25: ~8,000 tokens. * At Problem Size ~30: ~10,000 tokens. * At Problem Size ~35: ~12,000 tokens. * At Problem Size ~42: ~15,000 tokens. * At Problem Size ~50: ~18,000 tokens and ~20,000 tokens. * At Problem Size ~55: ~20,000 tokens and ~24,000 tokens. * At Problem Size ~64: ~22,000 tokens, ~28,000 tokens, and ~35,000 tokens. * At Problem Size ~72: ~20,000 tokens, ~31,000 tokens, and ~51,000 tokens (a very high point). * At Problem Size ~80: ~28,000 tokens, ~48,000 tokens. * At Problem Size ~100: ~41,000 tokens, ~44,000 tokens, and ~56,000 tokens (the highest point on the chart). * **Fit Line:** The dash-dot green regression line starts near (18, 5000) and ends near (100, 46000). The R-squared value of 0.813 indicates a strong fit, similar to the "medium" category. ### Key Observations 1. **Clear Hierarchy:** For any given Problem Size, the number of Reasoning Tokens increases systematically from Low to Medium to High effort. 2. **Increasing Slope with Effort:** The slope of the regression line becomes progressively steeper from Low to Medium to High effort, indicating that the *rate* at which token usage grows with problem size is greater for higher reasoning efforts. 3. **Variance Increases with Effort:** The scatter (vertical spread) of data points around the fit line is smallest for "low" effort and largest for "high" effort, particularly at larger problem sizes (e.g., Problem Size 72 and 100). 4. **Strong Correlation for Medium/High:** Both the "medium" and "high" effort categories show a strong linear correlation (R² > 0.8) between problem size and token usage. 5. **Potential Outliers:** The data point at Problem Size ~72 for "medium" effort (~23,000 tokens) appears high relative to its trend. The "high" effort series has several points at large problem sizes (e.g., ~51,000 at PS 72, ~56,000 at PS 100) that are significantly above the fit line. ### Interpretation The chart demonstrates a fundamental trade-off in computational reasoning: **increased problem complexity (size) requires more processing resources (tokens), and this cost scales more aggressively when the system is configured for higher reasoning effort.** * **Low Effort** appears to be a "baseline" mode where token usage grows slowly and unpredictably with problem size (low R²). It may represent a shallow or heuristic-based approach. * **Medium and High Effort** modes show a predictable, linear scaling law. The strong R² values suggest these modes engage in a more systematic, depth-first reasoning process whose resource consumption can be reliably modeled. * The **steeper slope for High Effort** implies that for complex problems, choosing a high-effort strategy incurs a multiplicative cost in tokens. This could be due to more extensive search, verification, or step-by-step explanation generation. * The **increased variance at High Effort** for large problems suggests that the reasoning process becomes less uniform; some problems may trigger exceptionally long chains of thought, while others of similar size are solved more efficiently. This could reflect the inherent variability in problem difficulty beyond just the "size" metric. In essence, the data suggests that "Reasoning Effort" is a critical control parameter that not only determines the absolute resource cost but also fundamentally changes how that cost scales with problem difficulty. Users or systems must balance the desire for thoroughness (high effort) against the predictable and potentially prohibitive increase in token consumption for large-scale tasks.

## Line Chart: Reasoning Tokens vs Problem Size ### Overview The chart illustrates the relationship between problem size (x-axis) and reasoning tokens consumed (y-axis) across three reasoning effort levels: low, medium, and high. Three data series with distinct markers and trend lines are plotted, each with associated R² values indicating model fit quality. ### Components/Axes - **X-axis (Problem Size)**: Ranges from 20 to 100 in increments of 20 - **Y-axis (Reasoning Tokens)**: Ranges from 0 to 50,000 in increments of 10,000 - **Legend**: Positioned in top-left corner with three entries: - Low (blue circles, solid line, R²=0.489) - Medium (orange squares, dashed line, R²=0.833) - High (green triangles, dotted line, R²=0.813) ### Detailed Analysis 1. **Low Effort (Blue Circles)**: - Data points cluster tightly around a shallow upward slope - Starts near 1,000 tokens at problem size 20 - Reaches ~4,000 tokens at problem size 40 - R²=0.489 indicates moderate linear correlation 2. **Medium Effort (Orange Squares)**: - Stronger upward trajectory than low effort - Begins at ~3,000 tokens at problem size 20 - Reaches ~18,000 tokens at problem size 80 - R²=0.833 shows excellent linear fit 3. **High Effort (Green Triangles)**: - Steepest slope among all series - Starts at ~5,000 tokens at problem size 20 - Peaks at ~45,000 tokens at problem size 100 - R²=0.813 indicates strong linear relationship - Notable outliers: 3 data points exceed trend line at problem sizes 60-100 ### Key Observations - All series show positive correlation between problem size and token consumption - High effort demonstrates 11x greater token usage than low effort at maximum problem size - Medium effort achieves best predictive accuracy (highest R²) - High effort series contains 3 outliers above predicted values at larger problem sizes - Low effort shows weakest linear relationship (lowest R²) ### Interpretation The data suggests that increased reasoning effort correlates with exponentially higher computational resource requirements. While all effort levels show linear scaling with problem size, the medium effort achieves optimal balance between predictive accuracy (R²=0.833) and resource efficiency. The high effort's outliers at larger problem sizes may indicate edge cases requiring disproportionate resources, potentially highlighting limitations in current reasoning architectures. These findings could inform AI system design by quantifying the trade-off between reasoning depth and computational cost, particularly for large-scale problem solving applications.