Problem Statement
Find all solutions where:
- A 2-digit number multiplied by a 3-digit number equals a 5-digit number
- All digits 0-9 are used exactly once across all three numbers
- The first digit of the 5-digit result is 1
Solutions Found
The solver found 6 valid solutions:
| Solution | 2-digit | 3-digit | 5-digit Result | Verification |
|---|---|---|---|---|
| 1 | 27 | 594 | 16038 | 27 × 594 = 16038 |
| 2 | 36 | 495 | 17820 | 36 × 495 = 17820 |
| 3 | 39 | 402 | 15678 | 39 × 402 = 15678 |
| 4 | 45 | 396 | 17820 | 45 × 396 = 17820 |
| 5 | 52 | 367 | 19084 | 52 × 367 = 19084 |
| 6 | 54 | 297 | 16038 | 54 × 297 = 16038 |
Verification
Each solution satisfies all constraints:
Constraint 1: Correct Multiplication
All multiplications are verified to produce the correct 5-digit result.
Constraint 2: Unique Digits (0-9)
Each solution uses all digits 0-9 exactly once across the three numbers:
Example - Solution 1 (27 × 594 = 16038):
- Digits used: 2, 7, 5, 9, 4, 1, 6, 0, 3, 8
- All digits 0-9 present: ✅
Constraint 3: 5-digit Result Starting with 1
All results are 5-digit numbers in the range 15,000-19,999:
- 16038 ✅
- 17820 ✅
- 15678 ✅
- 19084 ✅
Key Observations
Multiple Solutions
- The puzzle has 6 valid solutions, not just one
- This demonstrates the richness of the constraint space
Duplicate Results
- 16038 appears twice (solutions 1 and 6)
- 17820 appears twice (solutions 2 and 4)
- This shows different factor pairs can produce the same result
Range Analysis
- 2-digit numbers: Range from 27 to 54
- 3-digit numbers: Range from 297 to 594
- 5-digit results: Range from 15,678 to 19,084
Mathematical Properties
- All 5-digit results are in the 15,000-19,000 range
- The smallest 2-digit number used is 27
- The largest 2-digit number used is 54
- Results are well-distributed across the valid range
Algorithm
The solver uses a brute-force approach:
- Iterate through all 2-digit numbers (10-99)
- Iterate through all 3-digit numbers (100-999)
- Calculate the product
- Check if product is 5-digit and starts with 1
- Verify all digits 0-9 are used exactly once
- Collect valid solutions
Files
solve_puzzle.py- Main solver programPUZZLE.md- This documentation
Running the Solver
python3 solve_puzzle.pyConclusion
The conjecture that the first digit of the 5-digit number is 1 was correct. The puzzle has multiple valid solutions, each demonstrating the elegant constraint satisfaction where all digits 0-9 are used exactly once across the multiplication problem.
