In this number analogy, 2 is related to 16 by a power based rule; using the same rule, choose the number that correctly completes the pattern: 2 : 16 :: 3 : ?

Introduction / Context:
This problem is a classic number analogy question. The first pair, 2 and 16, follows a mathematical relationship that you must discover. Then you apply the same relationship to the second number, 3, in order to find the missing term. These questions are designed to test your familiarity with exponents, simple operations and pattern recognition in numbers.

Given Data / Assumptions:

• The first pair is 2 and 16.
• The second pair starts with 3 and the unknown number to be found.
• The same numeric operation must link 2 to 16 and 3 to the correct option.
• Only one option will match the rule consistently.

Concept / Approach:
When a small number maps to a considerably larger one, powers and exponents are often involved. We test whether 16 has a simple expression in terms of 2, such as 2 squared, 2 cubed or higher powers. Once we identify that 16 is 2 raised to a certain power, we apply that same exponent to 3 and then match the result with the available options.

Step-by-Step Solution:
Step 1: Express 16 in terms of 2. We know that 2^2 = 4, 2^3 = 8 and 2^4 = 16.Step 2: From the above, we see that 16 equals 2 raised to the power 4. So the rule connecting the first pair is n mapped to n^4.Step 3: Apply the same rule to 3. We must compute 3^4.Step 4: Calculate 3^2 = 9 and then 3^4 = 9^2 = 81.Step 5: Therefore, under the rule n mapped to n^4, the number corresponding to 3 is 81.Step 6: Check the options and note that 81 is given as option B.

Verification / Alternative check:
To verify, test whether any other simple pattern could connect 2 to 16 and 3 to one of the other options. For example, 2 multiplied by 8 gives 16, but 3 multiplied by 8 would give 24, which is not among the options. Similarly, 2 cubed plus something does not give a neat symmetric rule. The exponent pattern 2^4 and 3^4 is the simplest consistent relationship that explains the given pair and produces a listed option.

Why Other Options Are Wrong:
Option A, 340, does not correspond to any simple power of 3. Option C, 243, equals 3^5 and would require changing the exponent between pairs, which is not allowed. Option D, 122, does not have a direct power based connection with 3. Thus, they do not follow the same rule that maps 2 to 16.

Common Pitfalls:
Many students quickly see that 16 is a power of 2 but then incorrectly choose 243 as 3^5, thinking that higher exponents are better. Others try to use addition and multiplication combinations rather than the cleaner exponent rule. Always prioritise the simplest pattern that works perfectly for the first pair and then apply it unchanged to the second pair.

Final Answer:
Using the same rule, 3 is related to 81, so option B is correct.