In a certain number pattern, pairs of outer numbers are written with a middle number in brackets as follows: 14 (16) 18 and 33 (64) 25. Using the same rule, 25 (49) A is also written in this form. What is the correct value of A that fits this pattern?

Introduction:
This problem features a numerical pattern where two outer numbers are linked to a middle number written in brackets. The given examples are 14 (16) 18 and 33 (64) 25. We must observe the relationship among the three numbers in each case and then apply the same rule to determine the possible value or values of A in 25 (49) A. The question tests pattern discovery and careful handling of absolute differences.

Given Data / Assumptions:
The pairs and middle numbers given are:
1) 14 (16) 18
2) 33 (64) 25
We are told that 25 (49) A follows the same pattern. We assume:
1) The middle number in brackets is determined solely by the two outer numbers.
2) The same rule applies consistently to both examples and to the unknown case.

Concept / Approach:
On inspection, 16 and 64 in the middle look like perfect squares: 16 is 4^2 and 64 is 8^2. This suggests that the middle number might be the square of the difference between the outer numbers. We check whether this rule holds for both examples. If it does, then in 25 (49) A, the middle number 49 should similarly be the square of the absolute difference between 25 and A.

Step-by-Step Solution:
Step 1: Examine 14 (16) 18. Compute the difference of the outer numbers: |18 - 14| = 4. The square of this difference is 4^2 = 16, which matches the middle number.Step 2: Examine 33 (64) 25. Compute the difference: |33 - 25| = 8. The square of this difference is 8^2 = 64, which again matches the middle number.Step 3: The confirmed rule is: for outer numbers X and Y, the middle number equals (|X - Y|)^2.Step 4: Apply this to 25 (49) A. Here the middle number 49 must equal the square of the absolute difference between 25 and A.Step 5: So we have (|25 - A|)^2 = 49. Taking the positive square root gives |25 - A| = 7.Step 6: Solve the absolute value equation: 25 - A = 7 or 25 - A = -7.Step 7: From 25 - A = 7, we get A = 18. From 25 - A = -7, we get 25 + 7 = A, so A = 32.Step 8: Therefore, both A = 18 and A = 32 satisfy the pattern, meaning either value is mathematically valid.

Verification / Alternative check:
Check both candidates explicitly. For A = 18, the pair 25 and 18 yields |25 - 18| = 7, and 7^2 = 49. For A = 32, the pair 25 and 32 also yields |25 - 32| = 7, and again 7^2 = 49. In each case, the middle number remains 49 and the same rule holds, so both outer values are acceptable within this pattern.

Why Other Options Are Wrong:
The values 24 and 14 do not satisfy the equation (|25 - A|)^2 = 49, because |25 - 24| = 1 and |25 - 14| = 11, whose squares are 1 and 121, not 49. Choosing only 32 or only 18 would ignore the fact that the difference is taken in absolute value, allowing two symmetric solutions. The only option that correctly reflects all valid values is the combined choice "32 or 18".

Common Pitfalls:
Some learners forget that the absolute difference between two numbers can be realised in two ways, leading to two distinct outer values that fit the same square. Others may recognise only one solution and ignore the mirror case. It is important to treat |X - Y| as a distance, not an ordered subtraction, and to solve for all possible values that satisfy the relationship.

Final Answer:
Both A = 32 and A = 18 satisfy the given pattern, so the correct choice is 32 or 18.