In a certain numerical code, the symbol "@" denotes a special operation between two numbers. It is known that 5 @ 1 = 60, 12 @ 8 = 200 and 16 @ 2 = 180. Using the same rule, what is the value of 16 @ 10?

Introduction:
This question introduces a custom numeric operator "@". Instead of having a standard mathematical meaning, the operator is defined implicitly by several example equations. By analysing those examples, we must infer the general rule for a @ b and then apply that rule to compute 16 @ 10. It is a straightforward test of pattern discovery in number puzzles.

Given Data / Assumptions:
The three given relationships are:
1) 5 @ 1 = 60
2) 12 @ 8 = 200
3) 16 @ 2 = 180
We must find 16 @ 10. We assume:
1) The operation @ is defined consistently across all these equations.
2) The result is obtained through simple arithmetic combinations involving the sum of the two numbers, possibly scaled by a constant.

Concept / Approach:
Since the results 60, 200, and 180 end with a zero or are round numbers, a natural guess is that the operation involves multiplying a sum by 10 or by another fixed constant. We test whether (a + b) * 10 matches each given example. If this rule works for all, we can confidently apply it to 16 and 10.

Step-by-Step Solution:
Step 1: Test the first example: 5 @ 1. Compute (5 + 1) * 10.Step 2: The sum is 6, and 6 * 10 = 60, matching the given value.Step 3: Test the second example: 12 @ 8. Compute (12 + 8) * 10.Step 4: The sum is 20, and 20 * 10 = 200, again matching the given value.Step 5: Test the third example: 16 @ 2. Compute (16 + 2) * 10 = 18 * 10 = 180, which also matches.Step 6: Since the pattern (a + b) * 10 fits all three equations, we accept this as the definition of a @ b.Step 7: Apply the rule to 16 @ 10: compute (16 + 10) * 10.Step 8: The sum is 26, and 26 * 10 = 260.Step 9: Therefore, 16 @ 10 equals 260.

Verification / Alternative check:
We can consider other simple possibilities, such as a * b, a * b + 10, or (a - b) * 10, but none of these produce 60, 200, and 180 together from the given input pairs. Only the rule (a + b) * 10 matches all three cases simultaneously, so it is the most credible and consistent interpretation.

Why Other Options Are Wrong:
The values 32, 11, and 9 are much smaller and cannot be obtained by the discovered rule. The value 180 corresponds to 16 @ 2 and not to 16 @ 10. Thus, 260 is the only option consistent with the pattern already confirmed by the examples.

Common Pitfalls:
A frequent mistake is to fixate on multiplication of the two numbers and add or subtract arbitrary constants until the first example fits, without checking the subsequent examples. Another error is to overlook the common factor of 10 in the answers, which is a strong clue towards (a + b) * 10. Always ensure that the same simple rule holds for all given equations before using it on the unknown case.

Final Answer:
Using the established pattern, the value of 16 @ 10 is 260.