In a certain operator puzzle, the operation "α" is defined so that 16 α 1 = 8, 14 α 6 = 42 and 12 α 5 = 30. Using the same pattern, what is the value of 2 α 6?

Introduction / Context:
This question introduces a special operator "α" defined by examples. Your task is to deduce the hidden rule that links the pair of numbers on the left to the result on the right and then apply this rule to a new pair. Such problems check pattern recognition skills and speed in working with basic arithmetic operations.

Given Data / Assumptions:

• 16 α 1 = 8.
• 14 α 6 = 42.
• 12 α 5 = 30.
• We must compute 2 α 6.
• The same rule is used in all three examples.

Concept / Approach:
Look for a simple arithmetic relation involving the two numbers. Because the result in each case seems smaller than the product but larger than either number alone, dividing the product by a fixed constant is a strong possibility. Testing a formula like a α b = a * b / 2 is very natural here because products such as 16 * 1, 14 * 6 and 12 * 5 all give even numbers that are divisible by 2.

Step-by-Step Solution:
Step 1: For 16 α 1, compute (16 * 1) / 2 = 16 / 2 = 8, which matches the given result.Step 2: For 14 α 6, compute (14 * 6) / 2 = 84 / 2 = 42, again matching the given value.Step 3: For 12 α 5, compute (12 * 5) / 2 = 60 / 2 = 30, which also matches the given result.Step 4: Therefore the rule is a α b = (a * b) / 2.Step 5: Apply this rule to 2 α 6: (2 * 6) / 2 = 12 / 2 = 6.

Verification / Alternative check:
If we had tried a α b = a + b or a - b or a * b, one of the given examples would not match. The fact that (a * b) / 2 fits all three examples perfectly shows that it is the intended definition of α. Computing 2 α 6 with this rule gives 6, and no other simple consistent rule would produce a different result without breaking an earlier equality.

Why Other Options Are Wrong:

• 18 would come from using a * b / (some other divisor), which would not fit all examples.
• 4, 8 and 10 correspond to incorrect formulas such as a + b, b - a or half of the sum, none of which match the given three equations.

Common Pitfalls:
Candidates sometimes try to force elaborate expressions instead of testing simple possibilities like half the product. Another risk is using a different rule for each example. In these questions, there is always a single consistent pattern, so every guessed rule must be validated across all given equalities.

Final Answer:
Using the rule a α b = (a * b) / 2, we get 2 α 6 = 6.