## Line Chart: Multiplication Accuracy vs (Binned) Length of Thoughts
### Overview
The image presents three line charts comparing multiplication accuracy against the number of tokens, grouped by the size of the numbers being multiplied. The charts are divided into "Small Numbers" (1x1 to 6x6), "Medium Numbers" (7x7 to 11x11), and "Large Numbers" (12x12 to 20x20). Each chart displays the accuracy percentage on the y-axis and the number of tokens on the x-axis. Multiple lines represent different multiplication problems within each size category.
### Components/Axes
* **Title:** Multiplication Accuracy vs (Binned) Length of Thoughts
* **X-axis (Horizontal):** Number of Tokens, with markers at 1K, 4K, 8K, 11K, and 14K.
* **Y-axis (Vertical):** Accuracy (%), with markers at 0, 20, 40, 60, 80, and 100.
* **Chart Subtitles:**
* Small Numbers (1x1 to 6x6) - Located at the top-left.
* Medium Numbers (7x7 to 11x11) - Located at the top-center.
* Large Numbers (12x12 to 20x20) - Located at the top-right.
* **Legend:** Located at the bottom, mapping line colors and markers to specific multiplication problems (e.g., 1x1, 2x2, 3x3, ..., 20x20).
### Detailed Analysis
**1. Small Numbers (1x1 to 6x6)**
* The accuracy for small numbers is consistently high across all token counts.
* **1x1 (Dark Blue, Circle):** Starts at approximately 100% accuracy at 1K tokens and remains at 100% through 14K tokens.
* **2x2 (Dark Blue, Square):** Starts at approximately 100% accuracy at 1K tokens and remains at 100% through 14K tokens.
* **3x3 (Teal, Triangle):** Starts at approximately 100% accuracy at 1K tokens and remains at 100% through 14K tokens.
* **4x4 (Blue, Diamond):** Starts at approximately 100% accuracy at 1K tokens and remains at 100% through 14K tokens.
* **5x5 (Teal, Down Triangle):** Starts at approximately 100% accuracy at 1K tokens and remains at 100% through 14K tokens.
* **6x6 (Blue, Left Triangle):** Starts at approximately 100% accuracy at 1K tokens and remains at 100% through 14K tokens.
**2. Medium Numbers (7x7 to 11x11)**
* The accuracy for medium numbers varies more significantly with token count.
* **7x7 (Dark Blue, Circle):** Starts at approximately 25% accuracy at 1K tokens, rises to approximately 80% at 4K tokens, and declines to approximately 20% at 14K tokens.
* **8x8 (Dark Blue, Square):** Starts at approximately 95% accuracy at 1K tokens, reaches approximately 98% at 4K tokens, and declines to approximately 35% at 14K tokens.
* **9x9 (Teal, Triangle):** Starts at approximately 90% accuracy at 1K tokens, reaches approximately 95% at 4K tokens, and declines to approximately 90% at 14K tokens.
* **10x10 (Blue, Diamond):** Starts at approximately 70% accuracy at 4K tokens, reaches approximately 70% at 8K tokens, and declines to approximately 55% at 14K tokens.
* **11x11 (Teal, Down Triangle):** Starts at approximately 40% accuracy at 4K tokens, reaches approximately 50% at 8K tokens, and declines to approximately 10% at 14K tokens.
**3. Large Numbers (12x12 to 20x20)**
* The accuracy for large numbers is generally low across all token counts.
* **12x12 (Dark Blue, Circle):** Starts at approximately 0% accuracy at 1K tokens, reaches approximately 10% at 8K tokens, and declines to approximately 0% at 14K tokens.
* **13x13 (Dark Blue, Square):** Starts at approximately 0% accuracy at 1K tokens, reaches approximately 10% at 8K tokens, and declines to approximately 0% at 14K tokens.
* **14x14 (Teal, Triangle):** Starts at approximately 0% accuracy at 1K tokens, reaches approximately 12% at 11K tokens, and declines to approximately 0% at 14K tokens.
* **15x15 (Blue, Diamond):** Starts at approximately 0% accuracy at 1K tokens and remains at 0% through 14K tokens.
* **16x16 (Teal, Down Triangle):** Starts at approximately 0% accuracy at 1K tokens and remains at 0% through 14K tokens.
* **17x17 (Dark Blue, Circle):** Starts at approximately 0% accuracy at 1K tokens and remains at 0% through 14K tokens.
* **18x18 (Dark Blue, Square):** Starts at approximately 0% accuracy at 1K tokens and remains at 0% through 14K tokens.
* **19x19 (Teal, Triangle):** Starts at approximately 0% accuracy at 1K tokens and remains at 0% through 14K tokens.
* **20x20 (Blue, Diamond):** Starts at approximately 0% accuracy at 1K tokens and remains at 0% through 14K tokens.
### Key Observations
* **Small Numbers:** Consistently high accuracy, suggesting the model performs well on simple multiplication problems regardless of the number of tokens.
* **Medium Numbers:** Accuracy varies, with some problems showing a peak in accuracy at around 4K-8K tokens before declining. This suggests an optimal token length for these problems.
* **Large Numbers:** Consistently low accuracy, indicating the model struggles with more complex multiplication problems, regardless of the number of tokens.
* **Token Count Impact:** For medium numbers, there appears to be an optimal token count range (4K-8K) where accuracy is highest. Beyond this range, accuracy tends to decrease.
### Interpretation
The data suggests that the model's ability to perform multiplication accurately is highly dependent on the size of the numbers involved. Small number multiplication is consistently accurate, indicating a strong grasp of basic arithmetic. Medium number multiplication shows a more nuanced relationship with token count, suggesting that there's an optimal "length of thought" for these problems. The decline in accuracy beyond this optimal range could be due to the model becoming over-saturated with information or losing focus. Large number multiplication consistently yields low accuracy, indicating a fundamental limitation in the model's ability to handle complex arithmetic, regardless of the number of tokens provided. This could be due to the model's architecture, training data, or the inherent complexity of the problems themselves.
