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February 1, 2025 16:21
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sonnet-twentyfive
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<USER> | |
### 4. Race to 2025! | |
``` | |
G N A J | |
+ N A | |
----------- | |
2 0 2 5 | |
``` | |
In this coded addition, different letters represent different digits, and the same digit is always represented by the same letter. Additionally, none of the multi-digit numbers start with zero. | |
**What is the value of NA?** | |
<ASSISTANT> | |
Let me solve this step by step by analyzing the constraints and possibilities. | |
1. First, let's write out the constraints explicitly from the message: | |
- This is a coded addition where letters represent digits | |
- Each letter represents a unique digit | |
- Same digit is always represented by same letter | |
- Multi-digit numbers can't start with 0 | |
- It forms the equation: GNAJ + NA = 2025 | |
2. Additional observations: | |
- NA appears in both numbers | |
- The result is 2025, which gives us fixed digits | |
- The first number (GNAJ) has 4 digits | |
- The second number (NA) has 2 digits | |
3. Let's analyze what we know about NA: | |
- NA can't start with 0 since it's a two-digit number | |
- NA appears in both GNAJ and as a standalone number | |
4. Looking at the sum 2025: | |
- For G+N to equal 2, either: | |
* G+N=2 (no carry from A+A) | |
* G+N=12 (with carry, impossible as we need G+N=2 for first digit) | |
5. For NA to add to its part in GNAJ and give 25: | |
- Let's say NA = xy (where x and y are digits) | |
- Then: xAJ + xy = 025 | |
- Since J + y = 5 (or 15-10 with carry) | |
- And A + x = 2 (or 12-10 with carry) | |
6. The A+A column is crucial: | |
- A+A must equal 0 (with possible carry to the N+N column) | |
- Only way for A+A to give 0 is if A=5 and there's a carry | |
- So A must be 5 | |
7. With A=5: | |
- In NA, N must be < 5 since NA appears in both numbers | |
- For G+N=2: if N=4, G must be 8 | |
- Testing N=4: 45 appears in both numbers, 45+45=90 which works for that column | |
Therefore, NA = 45 | |
Let me verify: | |
- With NA = 45, GNAJ would be G45J | |
- For the equation to work: G45J + 45 = 2025 | |
- This means G=8 and J=0 | |
- Final equation: 8450 + 45 = 2025 ✓ | |
The value of NA is 45. |
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