Key Concepts and Formulas

• Factorial: $n! = n \times (n-1) \times (n-2) \times ... \times 2 \times 1$
• The problem involves pattern recognition. We need to identify the relationship between the outer numbers and the inner number in each figure.

Step-by-Step Solution

Step 1: Analyze the given examples to find the pattern. The proposed pattern is: $\text{Inside number} = (\text{Greater number} - \text{Smaller number})^{(\text{Greater number} - \text{Smaller number})!}$

Step 2: Verify the pattern with the first example: Outer numbers are 2 and 1. The inside number is 1. $(2-1)^{(2-1)!} = (1)^{1!} = 1^1 = 1$ This matches the given inside number.

Step 3: Verify the pattern with the second example: Outer numbers are 12 and 8. The inside number is $4^{24}$. $(12-8)^{(12-8)!} = (4)^{4!} = 4^{4 \times 3 \times 2 \times 1} = 4^{24}$ This matches the given inside number.

Step 4: Verify the pattern with the third example: Outer numbers are 7 and 4. The inside number is $3^6$. $(7-4)^{(7-4)!} = (3)^{3!} = 3^{3 \times 2 \times 1} = 3^6$ This matches the given inside number.

Step 5: Apply the pattern to find the missing value. Outer numbers are 5 and 3. $\text{Missing value} = (5-3)^{(5-3)!} = (2)^{(2)!} = 2^{2 \times 1} = 2^2 = 4$

Common Mistakes & Tips

• Carefully evaluate the factorial. $n!$ is often confused with $n$.
• Double-check the arithmetic operations to avoid simple errors.
• Always verify the pattern with all given examples before applying it to find the unknown value.

Summary

We identified a pattern relating the outer numbers to the inner number in the given figures. The pattern is that the inside number is equal to the difference of the outer numbers raised to the power of the factorial of the difference. We verified this pattern with all three given examples and then applied it to the figure with the missing value. The missing value is found to be 4.

Final Answer The final answer is $\boxed{4}$.