Pattern-based arithmetic puzzle: Given 16 - 2 = 2, 9 - 3 = 0, and 81 - 1 = 8 (using a hidden rule), determine the value of 64 - 4 under the same rule.

Introduction / Context:
Many verbal-reasoning and coding–decoding questions disguise a simple numerical transformation under arithmetic-looking expressions. The minus sign here is not ordinary subtraction; instead, each left part is mapped to the right result using a consistent hidden rule. Our job is to infer that rule from the three examples and apply it to 64 - 4.

Given Data / Assumptions:

• Samples: 16 - 2 = 2, 9 - 3 = 0, 81 - 1 = 8.
• Target: 64 - 4 = ?
• The symbol “-” likely represents a non-standard operation determined by a pattern.

Concept / Approach:
A quick way to test patterns is to check perfect squares and roots because 16, 9, and 81 are all perfect squares. Let f(x, y) be the hidden operation. Try f(x, y) = sqrt(x) - y. If the square root of x minus y matches each example, we have the rule.

Step-by-Step Solution:

Verification / Alternative check:
Consider other candidates like digit sums, logarithms, or factor counts; they will either fail one of the given instances or require ad hoc exceptions. The square-root rule works cleanly for all three examples and naturally anticipates a perfect square again (64).

Why Other Options Are Wrong:

• 2: Would imply sqrt(64) - 4 = 2, i.e., 8 - 4 = 2, which is false.
• 6: Would require 8 - 4 = 6, also false.
• 8: Confuses the rule with sqrt(x) itself; ignores the subtrahend.
• None of these: Incorrect because 4 fits perfectly.

Common Pitfalls:
Interpreting the minus sign literally or forcing a complicated rule (e.g., prime factors) when a simple, uniform transformation (square root then subtract) fits every exemplar without exception.

Final Answer:
4