In a certain numerical code, the operator "@" transforms two numbers as follows: 9@3 = 12, 15@4 = 22, and 16@14 = 4. Using the same rule, what is the value of 6@2?

Introduction / Context:
This question belongs to the coding decoding category for numbers. The operator "@" does not represent a standard arithmetic operation. Instead, it encodes a simple but hidden numerical relationship between its two arguments. Your task is to deduce this relationship from the given examples and then apply it to a new pair of numbers, 6 and 2.

Given Data / Assumptions:
- We have 9@3 = 12.
- We also have 15@4 = 22.
- We further have 16@14 = 4.
- We must compute 6@2 according to the same rule.
- The rule is assumed to be consistent across all these examples.

Concept / Approach:
A natural idea is to look at differences or sums between the two numbers, as these often produce simple patterns. Observing the examples suggests that the difference between the first and second numbers may be important. Therefore we experiment with expressions like a - b, 2 * (a - b), a + b, or combinations. We then check which formula exactly reproduces all given results.

Step-by-Step Solution:
Step 1: Examine 9@3 = 12. The difference 9 - 3 = 6, and 2 * (9 - 3) = 2 * 6 = 12, which matches the result. Step 2: Check the same rule on 15@4. Here 15 - 4 = 11, and 2 * (15 - 4) = 2 * 11 = 22, which matches. Step 3: Test the rule on 16@14. The difference 16 - 14 = 2, and 2 * (16 - 14) = 2 * 2 = 4, which again matches. Step 4: Therefore, the rule is a@b = 2 * (a - b). Step 5: Apply this rule to 6@2. Compute 6 - 2 = 4, then 2 * 4 = 8.

Verification / Alternative check:
The formula a@b = 2 * (a - b) fits all three given examples exactly, without any exceptions. Other candidate patterns, such as a + b or a - 2b, do not work consistently across all three equations. Because we have three independent confirmations, we can confidently use this rule to evaluate 6@2 as 8.

Why Other Options Are Wrong:
- Option 26 would require 2 * (6 - 2) or some other simple expression to equal 26, which is impossible without breaking the pattern seen in the examples.
- Option 1 does not arise from any natural expression such as difference or sum when we follow the established rule.
- Option 30 might come from multiplying 6 and 5 or misreading the operation, but it does not match the decoded rule.

Common Pitfalls:
Students sometimes guess a rule that fits only one or two of the given equations and move on without checking the remaining ones. Others focus only on sums and products, ignoring the possibility of scaled differences like 2 * (a - b). The safe technique is to systematically test each candidate rule against all examples before using it.

Final Answer:
Using the decoded pattern a@b = 2 * (a - b), the value of 6@2 is 8.