# Technical Document Extraction: Mathematical Problem Solving Approaches
## Section 1: Standard Prompting
### Model Input
```
User: 8 6 4 4
Assistant:
Answer: (4 + (8 - 6)) * 4 = 24.
User: 11 10 5 1
```
### Model Output
```
Assistant:
Answer: (11 - 1) * (10 - 5) = 24.
```
---
## Section 2: Algorithm of Thoughts
### Model Input
```
User: 8 6 4 4
Assistant:
1. 4 - 4: (left: 8, 6, 0)
...
- 4 + 2: (left: 6, 4) 10, 2, 24 = 6 * 4 -> found it!
Answer: (4 + (8 - 6)) * 4 = 24.
User: 11 10 5 1
```
### Model Output
```
Assistant:
1. 11 - 1: (left: 10, 10, 5)
...
- 12 * 10: (left: 120, 5) 24 = 120 / 5 -> found it!
Answer: ((11 + 1) * 10) / 5 = 24.
```
---
## Section 3: Chain-of-Thought
### Model Input
```
User: 8 6 4 4
Assistant:
8 - 6 = 2 (left: 4, 4, 2)
4 + 2 = 6 (left: 6, 4)
6 * 4 = 24
Answer: (4 + (8 - 6)) * 4 = 24.
User: 11 10 5 1
```
### Model Output
```
Assistant:
11 - 10 = 1 (left: 1, 1, 5)
1 * 1 = 1 (left: 1, 5)
5 * 1 = 5 (left: 5)
Answer: (((11 - 10) * 1) * 5) - 1 = 24.
```
---
## Section 4: Alternative Model Output
### Model Input
```
User: 8 6 4 4
Assistant:
8 - 6 = 2 (left: 4, 4, 2)
4 + 2 = 6 (left: 6, 4)
6 * 4 = 24
Answer: (4 + (8 - 6)) * 4 = 24.
User: 11 10 5 1
```
### Model Output
```
Assistant:
11 - 10 = 1 (left: 1, 1, 5)
1 * 1 = 1 (left: 1, 5)
5 * 1 = 5 (left: 5)
Answer: (((11 - 10) * 1) * 5) - 1 = 24.
```
---
## Key Observations
1. **Problem Structure**: All examples use four integers to reach a target value (24) through arithmetic operations.
2. **Approach Variations**:
- **Standard Prompting**: Direct answer format
- **Algorithm of Thoughts**: Step-by-step exploration with backtracking
- **Chain-of-Thought**: Sequential reasoning with intermediate steps
3. **Operational Patterns**:
- Common operations: subtraction, addition, multiplication, division
- Parenthetical grouping for operation precedence
- Explicit "left" number tracking in Algorithm of Thoughts
4. **Solution Diversity**: Multiple valid paths to the same result (e.g., 24) demonstrate algorithmic flexibility.
## Data Table Reconstruction
| Approach | Input 1 | Input 2 | Final Answer Format |
|------------------------|------------------|------------------|-----------------------------------------|
| Standard Prompting | 8 6 4 4 | 11 10 5 1 | (4 + (8-6)) * 4 = 24 |
| Algorithm of Thoughts | 8 6 4 4 | 11 10 5 1 | ((11+1)*10)/5 = 24 |
| Chain-of-Thought | 8 6 4 4 | 11 10 5 1 | (4 + (8-6)) * 4 = 24 |
## Spatial Grounding
- **Legend**: Not applicable (no visual elements)
- **Text Placement**: All labels and content follow a top-to-bottom, left-to-right hierarchy
## Trend Verification
- No numerical trends present (text-based reasoning only)
- All solutions converge to target value 24 through different operational paths
## Component Isolation
1. **Header**: Section titles (Standard Prompting, Algorithm of Thoughts, etc.)
2. **Main Content**: Model Input/Output pairs with reasoning traces
3. **Footer**: No footer content present
## Critical Notes
- All text is in English
- No numerical data tables present beyond the extracted examples
- No visual elements (charts/diagrams) detected
- All operations follow standard arithmetic rules with explicit precedence handling
