In a certain numerical code, the operation "*" between two numbers is defined so that 8*2 = 3, 14*18 = -2 and 18*2 = 8. Using the same rule, find the value of 14*10.

Introduction / Context: In this coding decoding problem, the symbol "*" represents a special numerical operation that combines two numbers in a nonstandard way. The examples 8*2 = 3, 14*18 = -2, and 18*2 = 8 all follow the same hidden rule. You must decode that rule and then apply it to the new expression 14*10 to find the correct value.

Given Data / Assumptions: - 8*2 = 3.
- 14*18 = -2.
- 18*2 = 8.
- We must evaluate 14*10 using the same operation.
- The rule is algebraic and consistent for all given examples.

Concept / Approach: Observing that some outputs are positive and some negative suggests that the operation might involve differences between the two numbers, possibly divided by a constant factor. A natural attempt is to compute (a - b) or (b - a) and then see whether halving the result matches the given outputs. Once a simple expression is found that fits all three cases, we can apply it confidently to 14 and 10.

Step-by-Step Solution: Step 1: Test the expression (a - b) / 2 for 8 and 2. Compute (8 - 2) / 2 = 6 / 2 = 3, which matches 8*2 = 3. Step 2: Test the same expression for 14 and 18. Compute (14 - 18) / 2 = (-4) / 2 = -2, which matches 14*18 = -2. Step 3: Test it for 18 and 2. Compute (18 - 2) / 2 = 16 / 2 = 8, which matches 18*2 = 8. Step 4: Therefore the rule is a*b = (a - b) / 2. Step 5: Apply this rule to 14 and 10: (14 - 10) / 2 = 4 / 2 = 2.

Verification / Alternative check: Having three consistent examples strongly supports the formula a*b = (a - b) / 2. If we attempted other expressions such as (b - a) / 2, (a + b) / 2, or a + b, they would not simultaneously reproduce all three given results. Recomputing each example confirms that we have the correct and unique simple rule. Therefore, using it for 14*10 is appropriate.

Why Other Options Are Wrong: - Option -5 would require (14 - 10) / 2 to be -5 or a different rule altogether, which breaks consistency with earlier equations.
- Option -2 corresponds to the specific case (14 - 18) / 2, not to 14 and 10.
- Option 6 could result from 14 - 8 or 3 * 2 in a mistaken pattern, but it does not align with the established formula.

Common Pitfalls: A frequent mistake is to focus on sums or products rather than differences when negative outputs are involved. Others may try several unrelated rules for different examples instead of searching for a single formula that fits all. The reliable strategy is to look for patterns that work uniformly across every given example and only then apply them to new inputs.

Final Answer: Using the decoded rule a*b = (a - b) / 2, the value of 14*10 is 2.