Understanding the Number Analogy Problem

The question asks us to identify a set of numbers from the given options that shares the same relationship between its elements as the relationships found in the two example sets: (2, 4, 18) and (1, 9, 50).

We are told that operations should be performed on the whole numbers provided, not on their individual digits.

Analyzing the Pattern in the Given Sets

Let's examine the first set: (2, 4, 18). Let the first number be $A=2$, the second number be $B=4$, and the third number be $C=18$. We need to find a rule that connects $A$, $B$, and $C$. The third number, 18, is significantly larger than 2 and 4, suggesting that multiplication, squares, or cubes might be involved in the pattern.

Let's try combining $A$ and $B$ using basic operations and squares:

• $A + B = 2 + 4 = 6$ (Not 18)
• $A \times B = 2 \times 4 = 8$ (Not 18)
• $A^2 = 2^2 = 4$, $B^2 = 4^2 = 16$
• $A^2 + B^2 = 4 + 16 = 20$ (Close to 18)
• $A + B^2 = 2 + 16 = 18$ (This matches the third number!)
• $A^2 + B = 4 + 4 = 8$ (Not 18)

The rule $C = A + B^2$ works for the first set (2, 4, 18) because $2 + 4^2 = 2 + 16 = 18$.

Now, let's test this rule with the second set: (1, 9, 50). Here, $A=1$, $B=9$, and $C=50$.

Using the rule $C = A + B^2$:

• $A + B^2 = 1 + 9^2 = 1 + 81 = 82$ (This is not 50).

So, the rule $C = A + B^2$ is not the correct pattern, as it only works for the first set.

Let's look for another pattern. What if we consider the sum of $A$ and $B$? For the first set, $A+B = 2+4 = 6$. If we square this sum, $(A+B)^2 = 6^2 = 36$. How is 36 related to 18? $36 / 2 = 18$.

Let's test the rule $C = \frac{(A+B)^2}{2}$ with the first set (2, 4, 18):

• $A=2, B=4, C=18$. $C = \frac{(2+4)^2}{2} = \frac{6^2}{2} = \frac{36}{2} = 18$. This works.

Now, let's test this rule with the second set (1, 9, 50):

• $A=1, B=9, C=50$. $C = \frac{(1+9)^2}{2} = \frac{10^2}{2} = \frac{100}{2} = 50$. This works!

The rule $C = \frac{(A+B)^2}{2}$ successfully describes the relationship in both given number sets. This rule involves whole number operations: addition, squaring, and division by 2, which results in a whole number $C$ because $(A+B)^2$ is even when $A$ and $B$ are integers (if $A$ and $B$ are both even, $A+B$ is even; if $A$ and $B$ are both odd, $A+B$ is even).

Applying the Rule to the Options

Now we will apply the rule $C = \frac{(A+B)^2}{2}$ to the options to find the set that follows the same pattern.

Identifying the Correct Set

From the table above, only Option 1: (3, 5, 32) follows the rule $C = \frac{(A+B)^2}{2}$. The calculation for this option yields 32, which matches the third number in the set.

Conclusion

The set (3, 5, 32) is related in the same way as the numbers of the sets (2, 4, 18) and (1, 9, 50). The pattern is that the third number is half the square of the sum of the first two numbers, i.e., $C = \frac{(A+B)^2}{2}$.

Revision Table: Key Concepts in Number Analogy

Additional Information: Mastering Number Series and Analogies

Solving number analogy and number series problems requires careful observation and practice. Here are some tips:

• Look for simple arithmetic relations first (addition, subtraction, multiplication, division).
• Consider squares and cubes of the numbers. Often, the third number is related to the square or cube of the first or second number, or a combination of them.
• Check for patterns involving the sum or difference of the first two numbers, squared or multiplied by a constant. In this problem, the pattern involved the square of the sum of the first two numbers.
• Look for relations between pairs of numbers within the set (e.g., A to B, B to C).
• Test one potential rule systematically across all given examples before applying it to the options.
• Be mindful of constraints mentioned in the question, like operating only on whole numbers.
• Practice different types of number puzzles to become familiar with common patterns.