Finding the Missing Number: Number Analogy 12 : 441

Analyze the Relationship

Examine the relationship between the first pair: 12 and 441.

Note that 441 is the square of 21, i.e., $441 = 21^2$.

The task is to find how 12 relates to 21.

Identify the Pattern Rule

Consider the digits of the first number (12): 1 and 2.

Calculate the absolute difference between the digits: $|1 - 2| = 1$.

Multiply this difference by 9: $1 \times 9 = 9$.

Add this result to the first number: $12 + 9 = 21$. This gives the root of the second number.

The pattern is: $N \rightarrow (N + 9 \times |d_1 - d_0|)^2$, where $N$ is the first number and $d_1, d_0$ are its digits.

Test the Pattern

Step 1: Verify with the first pair (12 : 441).

• Number $N = 12$. Digits are 1 and 2.
• Difference of digits: $|1 - 2| = 1$.
• Root calculation: $12 + (9 \times 1) = 12 + 9 = 21$.
• Result: $21^2 = 441$. This matches.

Step 2: Apply to the second pair (13 : ?).

• Number $N = 13$. Digits are 1 and 3.
• Difference of digits: $|1 - 3| = 2$.
• Root calculation: $13 + (9 \times 2) = 13 + 18 = 31$.
• Result: $31^2 = 961$.

Conclusion

The missing number is 961.