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Created February 1, 2025 16:21
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sonnet-twentyfive
<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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