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What number completes the pattern? Note that there is a single algebraic equation (limited to the basic arithmetic operations of addition, subtraction, multiplication, and division) that repeats across all rows. 6 

To solve the problem and find the number that completes the pattern, let's analyze the relationship between the numbers in each row.

### Row 1: \( 6 \, -5 \, -6 \, 5 \)
### Row 2: \( -4 \, 3 \, 2 \, -6 \)
### Row 3: \( 6 \, 6 \, 9 \, 4 \)
### Row 4: \( -9 \, ? \, 6 \, 3 \)

We need to identify a consistent arithmetic pattern across all rows.

1. **Row 1:**
   \[
   6 - 5 - 6 + 5 = 0
   \]
2. **Row 2:**
   \[
   -4 + 3 + 2 - 6 = -5
   \]
3. **Row 3:**
   \[
   6 + 6 - 9 + 4 = 7
   \]

It looks like the operations might differ between rows, so let's check for column-wise patterns instead:

### Column-wise Analysis:
1. **Column 1: \(6, -4, 6, -9\)**

   - The difference between each pair:
     - \(6 - (-4) = 10\)
     - \(-4 - 6 = -10\)
     - \(6 - (-9) = 15\)

2. **Column 2: \(-5, 3, 6, ?\)**

   - The differences:
     - \(-5 - 3 = -8\)
     - \(3 - 6 = -3\)
     - \(6 - ? = x\)

   If we expect a similar change, \(x = -3\). So, \(6 - (-3) = 9\). Therefore, \( ? = -3\).

3. **Column 3: \(-6, 2, 9, 6\)**

   - The differences:
     - \(-6 - 2 = -8\)
     - \(2 - 9 = -7\)
     - \(9 - 6 = 3\)

4. **Column 4: \(5, -6, 4, 3\)**

   - The differences:
     - \(5 - (-6) = 11\)
     - \(-6 - 4 = -10\)
     - \(4 - 3 = 1\)

### Conclusion:
Given this pattern of differences and focusing on the second column, the missing number in the fourth row should be \(-3\) to maintain the pattern of differences between each pair of numbers.

Thus, the missing number is **-3**.

Would you like more details on any specific part of this process? 

Here are some related questions:
1. How would the answer change if the operations between the numbers were different?
2. Could there be an alternative method to solve this type of problem?
3. How would you approach finding patterns in a different type of number grid?
4. What would happen if the number grid contained decimals or fractions instead of integers?
5. How does recognizing arithmetic patterns help in solving other mathematical problems?

**Tip:** Always check for patterns both row-wise and column-wise to identify any consistent operations or sequences.

Explore how to find the missing number in a patterned sequence using arithmetic operations. Detailed solution provided with step-by-step analysis of the relationship between numbers in each row and column. Suitable for Grades 7-9 students learning about arithmetic sequences and pattern recognition in math problems.

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