My 10-year-old niece sent me a puzzle that she wanted me to solve. She has her general knowledge paper tomorrow and was unable to get the correct answer, so she asked if I could solve the puzzle. My wife and I both tried to solve it, and we got the same answer that my niece already had. The problem was that, according to her book, that answer was not correct. So, I thought I’d ask reasoning models to find the answer.
Below is the puzzle. We have to find the value for the question mark:
It is a simple number pattern puzzle. You have to identify which pattern this puzzle uses to find the number. My wife, my niece, and I all got 9 as the answer. My niece’s book suggested 7 as the answer. We just couldn’t figure out how to get 7. To us, it looks like a printing mistake in her book.
Below is what each of the models produced. I uploaded the image and prompted them to “Solve the puzzle.”
xAI Grok 3
I had high hopes for Grok 3. It thought for more than 5 minutes and suggested that the final answer is 6. You can look at the complete conversation here: https://x.com/i/grok/share/Xzmb8WhIKFStucY4TrbzuXgft
It used differences between row pattern to arrive at 6 as the answer.
ChatGPT o3 mini
It thought for 1 minute and 50 seconds and produced 9 as the answer. It produced the same reasoning that we came up with as well.
A neat way to see the solution is by looking at the sums of each row:
- Row 1 (top): \(9\)
- Row 2: \(7 + 2 = 9\)
- Row 3: \(5 + 7 + 6 = 18\)
Notice that \(18\) is the sum of the two previous row-sums: \(9 + 9 = 18\).
If we assume this Fibonacci-like pattern continues, then Row 4 should be
\(\text{Row 3 sum} + \text{Row 2 sum} = 18 + 9 = 27.\)
We know the numbers in Row 4 are \(4, 6, \,?,\, 8\). Their sum must be 27:
\[
4 + 6 + ? + 8 = 27 \quad\Rightarrow\quad 18 + ? = 27 \quad\Rightarrow\quad ? = 9.
\]
Hence, the missing number is \(\boxed{9}.\)
Claude 3.7 Sonnet
I tried Claude 3.7 Sonnet with Extended Thinking last. It thought for 1 minute and 24 seconds and also produced 9 as the answer. If you look at the explanation, it doesn’t seem entirely right. According to its logic, the Row 4 sum should be 36, not 27. So, the final answer should be 18, not 9. Even though it got 9 as the answer, the reasoning looks flawed.
Looking at this triangular number pattern, I need to find what number belongs in place of the question mark.
I notice that if I examine the sum of each row:
- Row 1: 9
- Row 2: 7 + 2 = 9
- Row 3: 5 + 7 + 6 = 18
- Row 4: 4 + 6 + ? + 8 = ?
There appears to be a pattern where each row's sum equals 9 multiplied by the row number:
- Row 1: 9 × 1 = 9 ✓
- Row 2: 9 × 1 = 9 ✓
- Row 3: 9 × 2 = 18 ✓
- Row 4: 9 × 3 = 27
So for the fourth row, we need 4 + 6 + ? + 8 = 27
Therefore: ? = 27 - 4 - 6 - 8 = 27 - 18 = 9
The missing number is 9.
But book says answer is 7
The discrepancy with my niece’s book, which suggests 7 as the answer, remains unexplained. So again I asked LLMs to help me understand why answer 7.
Only ChatGPT with o3-mini got it right. It explained how 7 could be one of the possible answer. It thought for 2mins 53 seconds and gave below answer.
A common reason some books (or puzzle-setters) give “7” instead of “9” is that **they are using a “side‐sum” rule** rather than a row‐sum or Fibonacci‐type rule. In other words, they want **each of the three sides** of the triangle to add up to the same total. Here’s how that works:
---
## 1. Label the circles for clarity
Imagine the triangle like this:
```
(Top)
9
7 2
5 7 6
4 6 ? 8
```
We have three “sides” if we trace from the top circle down left, down right, and along the bottom:
1. **Left side**: 9 → 7 → 5 → 4
2. **Right side**: 9 → 2 → 6 → 8
3. **Bottom side**: 4 → 6 → ? → 8
---
## 2. Check the sums of the known sides
- **Left side sum**: \(9 + 7 + 5 + 4 = 25\)
- **Right side sum**: \(9 + 2 + 6 + 8 = 25\)
Both sides already total 25.
---
## 3. Impose the “all sides equal” rule
If the puzzle’s rule is that *all three sides must sum to the same total*, then the bottom side must also sum to 25:
\[
4 + 6 + (?) + 8 = 25.
\]
We already have \(4 + 6 + 8 = 18\). So
\[
18 + (?) = 25
\quad\Longrightarrow\quad
? = 7.
\]
Hence, **the missing number is 7** under the “equal side‐sum” pattern.
---
## 4. Why some solvers get 9 instead
A *different* common approach is to look at **row sums** (like a “Fibonacci‐style” growth of 9, 9, 18, and then 27). Under that method, you would set row 4 to sum to 27, which forces the missing circle to be 9. That is a *valid* pattern too, just a *different* puzzle rule.
**But** if the original puzzle (as in your book) explicitly says or implies “the sum of the numbers on each side of the triangle is the same,” then the correct solution is indeed **7**, not 9.
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