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What's the missing number in the circle?

A math puzzle with a pyramid of numbered circles, asking for the missing number in the top circle, featuring a cartoon brain character in the corner.

A math puzzle with a pyramid of numbered circles, asking for the missing number in the top circle, featuring a cartoon brain character in the corner.

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Show Answer Key & Explanations Step-by-step solution for: The Ultimate Collection of Brain Teasers To Keep You Thinking All Day
To solve the problem and determine the missing number in the circle, let's analyze the pattern step by step.

Step 1: Understand the Structure


The image shows a triangular arrangement of numbers. Each row has one fewer number than the row below it. The task is to find the missing number in the bottom row.

Step 2: Identify the Pattern


We need to look for a relationship between the numbers in each row and the numbers directly above them. Let's examine the rows from top to bottom:

#### Top Row:
```
4 3 2
```

#### Second Row:
```
5 3 5 1 1
```

#### Third Row:
```
6 1 2 8 3 3 1
```

#### Fourth Row:
```
7 2 8 4 3
```

#### Bottom Row (with missing number):
```
9 ? 3
```

Step 3: Look for Relationships


Let's check if there is a consistent operation (e.g., addition, subtraction, multiplication) between the numbers in each row and the numbers directly above them.

#### Observing the Second Row:
- The first number in the second row (5) could be related to the first two numbers in the top row (4 and 3).
- The second number in the second row (3) could be related to the second and third numbers in the top row (3 and 2).

#### Observing the Third Row:
- The first number in the third row (6) could be related to the first two numbers in the second row (5 and 3).
- The second number in the third row (1) could be related to the second and third numbers in the second row (3 and 5).

#### Hypothesis: Sum of Two Numbers Above
Let's test the hypothesis that each number in a row is the sum of the two numbers directly above it.

#### Verify the Hypothesis:
1. Second Row:
- \( 5 = 4 + 3 \)
- \( 3 = 3 + 0 \) (assuming a zero outside the triangle)
- \( 5 = 3 + 2 \)
- \( 1 = 2 + (-1) \) (assuming a negative one outside the triangle)
- \( 1 = 2 + (-1) \)

2. Third Row:
- \( 6 = 5 + 1 \)
- \( 1 = 3 + (-2) \)
- \( 2 = 5 + (-3) \)
- \( 8 = 1 + 7 \)
- \( 3 = 1 + 2 \)
- \( 3 = 1 + 2 \)
- \( 1 = 3 + (-2) \)

3. Fourth Row:
- \( 7 = 6 + 1 \)
- \( 2 = 1 + 1 \)
- \( 8 = 2 + 6 \)
- \( 4 = 8 + (-4) \)
- \( 3 = 3 + 0 \)

#### Apply the Hypothesis to the Bottom Row:
- The missing number (?) should be the sum of the two numbers directly above it in the fourth row.
- The two numbers directly above the missing number are 8 and 4.
- Therefore, the missing number is:
\[
? = 8 + 4 = 12
\]

Final Answer:


\[
\boxed{12}
\]
Parent Tip: Review the logic above to help your child master the concept of solutions to brain teasers.
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