## Heatmap: Performance Delta Across Models and Mathematical Domains ### Overview The image is a heatmap displaying the performance delta (in percentage) across different mathematical domains and puzzle types. The color intensity represents the magnitude of the performance difference, with green indicating positive delta (better performance) and red indicating negative delta (worse performance). ### Components/Axes * **Title:** Performance Delta Across Models and Mathematical Domains * **Y-axis (Rows):** Mathematical Domains: * Algebra and Number Theory * Analysis and Differential Equations * Applied and Computational Mathematics * Arithmetic * Foundations and Logic * Geometry and Topology * Probability, Statistics, and Discrete Mathematics * **X-axis (Columns):** Puzzle Types: * Sudoku * Nonogram * Cryptarithm * Magic Square * Zebra puzzle * Graph * Knight & Knaves * All-Game * **Color Legend (Right):** Delta from Base (%) * Dark Green: +4 * Light Green: +2 * Yellow: 0 * Orange: -2 * Dark Red: -4 ### Detailed Analysis The heatmap presents performance delta values for each combination of mathematical domain and puzzle type. The values are percentages, indicating the difference in performance compared to a baseline. * **Algebra and Number Theory:** Performance ranges from -0.78% to +1.48%. Cryptarithm shows the highest positive delta (+1.17%), while Zebra puzzle shows a negative delta of -0.78%. * **Analysis and Differential Equations:** Performance ranges from -0.42% to +4.61%. Magic Square shows the highest positive delta (+3.77%), and Sudoku shows a negative delta of -0.42%. * **Applied and Computational Mathematics:** Performance is consistently negative, ranging from -1.58% to -5.26%. Magic Square shows the most negative delta (-5.26%). * **Arithmetic:** Performance is generally slightly negative, ranging from -0.80% to +0.20%. Magic Square shows a slight positive delta (+0.20%). * **Foundations and Logic:** Performance is mostly negative (-5.88%) except for Cryptarithm and Zebra puzzle, and All-Game which show 0.00% delta. * **Geometry and Topology:** Performance is mostly positive, ranging from -0.86% to +2.73%. Nonogram shows the highest positive delta (+2.73%). * **Probability, Statistics, and Discrete Mathematics:** Performance is mixed, ranging from -1.08% to +1.22%. Nonogram shows the highest positive delta (+1.22%), and All-Game shows the most negative delta (-1.08%). ### Key Observations * Applied and Computational Mathematics and Foundations and Logic consistently show negative performance deltas across most puzzle types. * Analysis and Differential Equations shows the highest positive performance delta, particularly for Magic Square. * The performance varies significantly depending on the combination of mathematical domain and puzzle type. ### Interpretation The heatmap visualizes how different models perform across various mathematical domains when applied to different puzzle types. The data suggests that the effectiveness of a model is highly dependent on the specific combination of mathematical domain and puzzle type. For example, models perform poorly in Applied and Computational Mathematics, especially with Magic Square puzzles, while they perform well in Analysis and Differential Equations, again with Magic Square puzzles. This could indicate that certain models are better suited for specific types of problems or that certain mathematical domains are more challenging for the models in general. The negative performance in Foundations and Logic suggests potential limitations in handling logical reasoning tasks. The variations highlight the importance of selecting appropriate models based on the specific problem domain.

\n ## Heatmap: Performance Delta Across Models and Mathematical Domains ### Overview This heatmap visualizes the performance delta (percentage change) of a model across various mathematical domains and specific problem types within those domains. The color intensity represents the magnitude of the delta, with red indicating negative deltas (performance decrease) and green indicating positive deltas (performance increase). ### Components/Axes * **Title:** "Performance Delta Across Models and Mathematical Domains" (positioned at the top-center) * **Y-axis (Rows):** Mathematical Domains. The domains listed are: * Algebra and Number Theory * Analysis and Differential Equations * Applied and Computational Mathematics * Arithmetic * Foundations and Logic * Geometry and Topology * Probability, Statistics, and Discrete Mathematics * **X-axis (Columns):** Problem Types within the domains. The problem types listed are: * Sudoku * Nonogram * Cryptarithm * Magic Square * Zebra puzzle * Graph * Knight & Knaves * All-Game * **Color Scale (Legend):** Located on the right side of the heatmap. It represents the "Delta from Base (%)", ranging from approximately -4% to +4%. * Dark Red: Approximately -4% * Red: Approximately -2% * White: Approximately 0% * Light Green: Approximately +2% * Dark Green: Approximately +4% ### Detailed Analysis The heatmap contains 7 rows (mathematical domains) and 8 columns (problem types), resulting in 56 data points. The values are percentage changes, and are presented in the cells. Here's a breakdown of the data, row by row: * **Algebra and Number Theory:** * Sudoku: -0.31% * Nonogram: -0.62% * Cryptarithm: +1.17% * Magic Square: -0.16% * Zebra puzzle: -0.78% * Graph: +1.48% * Knight & Knaves: +0.16% * All-Game: +0.70% * **Analysis and Differential Equations:** * Sudoku: -0.42% * Nonogram: +0.00% * Cryptarithm: +1.83% * Magic Square: +3.77% * Zebra puzzle: +0.00% * Graph: +0.21% * Knight & Knaves: +1.46% * All-Game: +4.61% * **Applied and Computational Mathematics:** * Sudoku: -2.63% * Nonogram: -2.10% * Cryptarithm: -3.68% * Magic Square: -5.26% * Zebra puzzle: -3.16% * Graph: -2.10% * Knight & Knaves: -1.58% * All-Game: -1.58% * **Arithmetic:** * Sudoku: -0.65% * Nonogram: -0.80% * Cryptarithm: -0.57% * Magic Square: -0.20% * Zebra puzzle: -0.80% * Graph: -0.30% * Knight & Knaves: -0.27% * All-Game: -0.42% * **Foundations and Logic:** * Sudoku: -5.88% * Nonogram: -5.88% * Cryptarithm: -0.00% * Magic Square: -5.88% * Zebra puzzle: -0.00% * Graph: -5.88% * Knight & Knaves: -5.88% * All-Game: +0.00% * **Geometry and Topology:** * Sudoku: +1.29% * Nonogram: +2.73% * Cryptarithm: +1.29% * Magic Square: +0.72% * Zebra puzzle: +0.86% * Graph: +1.01% * Knight & Knaves: +0.29% * All-Game: -0.86% * **Probability, Statistics, and Discrete Mathematics:** * Sudoku: +0.68% * Nonogram: +1.22% * Cryptarithm: +0.95% * Magic Square: -0.14% * Zebra puzzle: +0.41% * Graph: -0.14% * Knight & Knaves: -0.27% * All-Game: -1.08% ### Key Observations * **Foundations and Logic** consistently shows the largest negative deltas across all problem types, except for "All-Game" which shows no change. * **Applied and Computational Mathematics** also exhibits largely negative deltas, with the most significant decrease observed for "Magic Square" (-5.26%). * **Analysis and Differential Equations** demonstrates the most positive deltas, particularly for "Magic Square" (+3.77%) and "All-Game" (+4.61%). * **Geometry and Topology** generally shows positive deltas, with "Nonogram" having the largest positive change (+2.73%). * The "All-Game" performance is mixed, showing no change for Foundations and Logic, a large positive change for Analysis and Differential Equations, and negative changes for Applied and Computational Mathematics and Probability, Statistics, and Discrete Mathematics. ### Interpretation The heatmap suggests that the model performs significantly worse on problems related to Foundations and Logic, and to a lesser extent, Applied and Computational Mathematics. Conversely, it performs better on problems within Analysis and Differential Equations and Geometry and Topology. The "All-Game" metric, representing a combined performance, shows a varied response, indicating that the model's strengths and weaknesses are not uniformly distributed across different problem types. The large negative deltas in Foundations and Logic could indicate a limitation in the model's ability to handle abstract reasoning or formal systems. The positive deltas in Analysis and Differential Equations might suggest the model is well-suited for tasks involving continuous functions and calculus. The mixed results for "All-Game" suggest that simply averaging performance across domains may not accurately reflect the model's overall capabilities. Further investigation is needed to understand the underlying reasons for these performance differences. This could involve analyzing the model's architecture, training data, and the specific characteristics of each problem type. The heatmap provides a valuable starting point for identifying areas where the model needs improvement and for guiding future research efforts.

## Heatmap: Performance Delta Across Models and Mathematical Domains ### Overview The image is a heatmap titled "Performance Delta Across Models and Mathematical Domains." It visualizes the percentage change in performance (delta from a base model) for eight different models or puzzle types across seven distinct mathematical domains. The data is presented in a grid where each cell's color and numerical value represent the performance delta. A color scale on the right provides a key for interpreting the values. ### Components/Axes * **Title:** "Performance Delta Across Models and Mathematical Domains" (centered at the top). * **Y-Axis (Rows - Mathematical Domains):** Seven categories are listed vertically on the left side: 1. Algebra and Number Theory 2. Analysis and Differential Equations 3. Applied and Computational Mathematics 4. Arithmetic 5. Foundations and Logic 6. Geometry and Topology 7. Probability, Statistics, and Discrete Mathematics * **X-Axis (Columns - Models/Puzzles):** Eight categories are listed horizontally at the bottom, rotated at a 45-degree angle: 1. Sudoku 2. Nonogram 3. Cryptarithm 4. Magic Square 5. Zebra puzzle 6. Graph 7. Knight & Knaves 8. All-Game * **Legend/Color Scale:** A vertical bar on the right side labeled "Delta from Base (%)". The scale ranges from approximately -5% (dark red) to +5% (dark green), with 0% represented by a pale yellow. Key markers are at -4, -2, 0, 2, and 4. * **Data Cells:** Each cell in the 7x8 grid contains a percentage value (e.g., "-0.31%", "+3.77%") and is colored according to the legend. ### Detailed Analysis The following table reconstructs the heatmap's data, with rows as Mathematical Domains and columns as Models/Puzzles. Values are the "Delta from Base (%)". | Mathematical Domain / Model | Sudoku | Nonogram | Cryptarithm | Magic Square | Zebra puzzle | Graph | Knight & Knaves | All-Game | | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | | **Algebra and Number Theory** | -0.31% | -0.62% | +1.17% | +0.16% | -0.78% | +1.48% | +0.16% | +0.70% | | **Analysis and Differential Equations** | -0.42% | +0.00% | +1.88% | +3.77% | +0.00% | +0.21% | +1.46% | +4.61% | | **Applied and Computational Mathematics** | -2.63% | -2.10% | -3.68% | -5.26% | -3.16% | -2.10% | -1.58% | -1.58% | | **Arithmetic** | -0.65% | -0.80% | -0.57% | +0.20% | -0.80% | -0.30% | -0.27% | -0.42% | | **Foundations and Logic** | -5.88% | -5.88% | +0.00% | -5.88% | +0.00% | -5.88% | -5.88% | +0.00% | | **Geometry and Topology** | +1.29% | +2.73% | +1.29% | +0.72% | +0.86% | +1.01% | +0.29% | -0.86% | | **Probability, Statistics, and Discrete Mathematics** | +0.68% | +1.22% | +0.95% | -0.14% | +0.41% | -0.14% | -0.27% | -1.08% | ### Key Observations 1. **Strongest Positive Performance:** The largest positive delta is **+4.61%** for the "All-Game" model in the "Analysis and Differential Equations" domain (dark green cell). The "Magic Square" model also shows strong positive performance in this domain (+3.77%). 2. **Strongest Negative Performance:** The most significant negative deltas are **-5.88%**, occurring multiple times in the "Foundations and Logic" domain for the Sudoku, Nonogram, Magic Square, Graph, and Knight & Knaves models (dark red cells). The "Magic Square" model also shows a large negative delta of **-5.26%** in "Applied and Computational Mathematics." 3. **Domain-Wide Trends:** * **Applied and Computational Mathematics:** Shows consistently negative deltas across all models, ranging from -1.58% to -5.26%. This domain appears to be the most challenging for all evaluated models relative to the base. * **Foundations and Logic:** Exhibits a polarized pattern. Performance is either unchanged (+0.00%) or significantly worse (-5.88%) across models, with no intermediate values. * **Analysis and Differential Equations:** Shows predominantly positive or neutral deltas, suggesting models generally perform better than the base in this domain. * **Geometry and Topology:** Shows mostly positive deltas, with the "Nonogram" model achieving the highest gain in this domain (+2.73%). 4. **Model-Specific Trends:** * **Magic Square:** Shows extreme variance, with the highest positive delta in one domain (+3.77% in Analysis) and the lowest negative delta in another (-5.26% in Applied Math). * **All-Game:** Shows the single highest positive delta (+4.61%) but also a notable negative delta in "Probability, Statistics, and Discrete Mathematics" (-1.08%). ### Interpretation This heatmap provides a comparative analysis of how different AI models (or puzzle-solving approaches) perform relative to a baseline across specialized mathematical fields. * **What the data suggests:** The performance delta is highly domain- and model-dependent. No single model excels universally. The "Analysis and Differential Equations" domain appears to be an area of relative strength for the tested models, while "Applied and Computational Mathematics" is a universal area of weakness. The stark, binary results in "Foundations and Logic" suggest a possible categorical failure or a specific benchmarking quirk for those models in that domain. * **How elements relate:** The color gradient allows for immediate visual identification of strengths (green) and weaknesses (red). The grid structure enables direct comparison: one can scan a row to see how a single model performs across all domains, or scan a column to see which models are best suited for a specific mathematical domain. * **Notable anomalies:** The repeated -5.88% value in "Foundations and Logic" is a significant outlier. Such a precise, repeated negative value could indicate a systematic error, a ceiling/floor effect in the benchmark, or a fundamental limitation of those models in handling logical foundations. The complete absence of positive deltas in "Applied and Computational Mathematics" is another critical finding, highlighting a major challenge area for current models.

## Heatmap: Performance Delta Across Models and Mathematical Domains ### Overview This heatmap visualizes performance deltas (percentage changes from a base value) across mathematical domains and puzzle types. The color gradient ranges from red (-5.88%) to green (+4.61%), with numerical values embedded in each cell. ### Components/Axes - **X-axis (Columns)**: Puzzle types: Sudoku, Nonogram, Cryptarithm, Magic Square, Zebra puzzle, Graph, Knight & Knaves, All-Game. - **Y-axis (Rows)**: Mathematical domains: Algebra and Number Theory, Analysis and Differential Equations, Applied and Computational Mathematics, Arithmetic, Foundations and Logic, Geometry and Topology, Probability, Statistics, and Discrete Mathematics. - **Legend**: Color scale from -5.88% (dark red) to +4.61% (dark green), with intermediate values in yellow/orange. ### Detailed Analysis - **Algebra and Number Theory**: - Strongest performance in Cryptarithm (+1.17%) and All-Game (+0.70%). - Weakest in Sudoku (-0.31%) and Knight & Knaves (-0.62%). - **Analysis and Differential Equations**: - Highest delta in Magic Square (+3.77%) and All-Game (+4.61%). - Neutral in Nonogram and Zebra puzzle (0.00%). - **Applied and Computational Mathematics**: - Consistently negative deltas (-2.63% to -1.58%) across all puzzles except Cryptarithm (-3.68%). - **Arithmetic**: - Mixed performance: Strong in Magic Square (+0.20%) but weak in Nonogram (-0.80%). - **Foundations and Logic**: - Dominantly negative (-5.88%) except in Cryptarithm and Zebra puzzle (0.00%). - **Geometry and Topology**: - Positive in Nonogram (+2.73%) and Cryptarithm (+1.29%), but negative in Zebra puzzle (-0.86%). - **Probability, Statistics, and Discrete Mathematics**: - Slightly positive in Nonogram (+1.22%) and Cryptarithm (+0.95%), but negative in Magic Square (-0.14%). ### Key Observations 1. **Highest Positive Delta**: Analysis and Differential Equations in Magic Square (+3.77%) and All-Game (+4.61%). 2. **Lowest Delta**: Foundations and Logic in Sudoku, Nonogram, Magic Square, Graph, and Knight & Knaves (-5.88%). 3. **Neutral Performance**: Foundations and Logic in Cryptarithm and Zebra puzzle (0.00%). 4. **Consistent Negatives**: Applied and Computational Mathematics underperforms across all puzzles. ### Interpretation - **Domain-Specific Strengths**: Analysis and Differential Equations excel in complex puzzles like Magic Square and All-Game, suggesting alignment with advanced problem-solving. Foundations and Logic’s poor performance in most puzzles may indicate a mismatch with their abstract nature. - **Puzzle-Type Trends**: Magic Square and All-Game show higher variability, possibly due to their reliance on diverse mathematical reasoning. - **Anomalies**: Foundations and Logic’s 0.00% delta in Cryptarithm and Zebra puzzle implies these puzzles may better suit their logical frameworks despite overall underperformance. - **Applied Mathematics Gap**: Negative deltas across all puzzles suggest these models struggle with the base performance benchmark, warranting further investigation into their design or training data. The data highlights the importance of domain-puzzle alignment, with certain mathematical disciplines outperforming others in specific problem types.