Understanding the Number Analogy: 256 : 290 :: 961 : ?

The question asks us to find the number that shares a similar relationship with 961 as 290 shares with 256. This is a common type of logical reasoning problem where we need to identify the pattern or rule connecting the first pair of numbers and apply it to the second number to find the missing term.

Analyzing the Relationship between 256 and 290

Let's look at the first pair of numbers: 256 and 290. We need to find a mathematical operation or pattern that transforms 256 into 290 or relates them in a specific way.

• We notice that 256 is a perfect square. Specifically, $\sqrt{256} = 16$, so $256 = 16^2$.
• Now let's see how 290 relates to 16 or 256.
• Consider the next integer after 16, which is 17.
• Let's calculate the square of 17: $17^2 = 17 \times 17 = 289$.
• How does 290 relate to 289? $290 = 289 + 1$.

So, the pattern appears to be: if the first number is $X^2$, the second number is $(X+1)^2 + 1$. Let's verify this rule with the first pair:

• First number: 256. Here, $X^2 = 256$, so $X = 16$.
• Applying the rule, the second number should be $(16+1)^2 + 1 = 17^2 + 1 = 289 + 1 = 290$.

The rule holds true for the pair 256 : 290.

Applying the Pattern to 961

Now we apply the same rule to the second part of the analogy, 961 : ?. The first number is 961.

• We need to find the number whose square is 961. We know that $30^2 = 900$ and $31^2 = 961$.
• So, for 961, $X^2 = 961$, which means $X = 31$.
• According to the pattern, the related number should be $(X+1)^2 + 1$.
• Substitute $X = 31$: $(31+1)^2 + 1 = 32^2 + 1$.
• Calculate $32^2$: $32 \times 32$.

  
  

  Calculation	Result
  $32 \times 32$	1024

• Now add 1 to the result: $1024 + 1 = 1025$.

Thus, the number related to 961 by the same pattern is 1025.

Comparing with the Options

The calculated number is 1025. Let's check the given options:

1. 1011
2. 1017
3. 1025
4. 1023

Our calculated value, 1025, matches option 3.

Conclusion

The relationship in the analogy 256 : 290 :: 961 : ? is that the second number is obtained by taking the square root of the first number (let's call it X), adding 1 to X, squaring the result, and finally adding 1. Applying this pattern to 961 (which is $31^2$) gives $(31+1)^2 + 1 = 32^2 + 1 = 1024 + 1 = 1025$.

Revision Table: Number Analogy Pattern

Additional Information: Solving Number Analogies

Number analogy questions test your ability to find patterns in numbers. Common patterns include:

• Arithmetic operations (addition, subtraction, multiplication, division)
• Squaring or cubing numbers, or taking their roots
• Patterns involving consecutive numbers
• Combinations of operations
• Prime numbers, composite numbers, etc.

To solve number analogies effectively, it is helpful to be familiar with squares and cubes of common numbers and practice identifying different types of relationships between numbers.