In the letter–number analogy “CD : 916 :: ____ : 100144”, which pair of letters follows the same positional square pattern?

Introduction / Context:
This analogy question combines alphabet positions with number patterns. The pair “CD : 916” suggests that the letters C and D are converted into the number 916 using a particular rule. You must discover this rule and then apply it to another pair of letters so that the result is 100144. This tests your ability to work with letter positions in the alphabet and to recognise patterns involving squares of numbers.

Given Data / Assumptions:

• First pair: CD and 916.
• Second pair: unknown letter pair and 100144.
• Alphabet positions: A = 1, B = 2, C = 3, D = 4, J = 10, L = 12, etc.
• Options: HG, MO, RS, JL, NP.

Concept / Approach:
First, we look at the letters in CD. C is the 3rd letter of the alphabet and D is the 4th. If we square these positions, we get 3^2 = 9 and 4^2 = 16. When we write these two results side by side, we get 916, which matches the given number for CD. The pattern is therefore “concatenate the squares of the alphabet positions of the two letters.” We now need two letters whose positions squared, and then written side by side, give 100144. Since 100 = 10^2 and 144 = 12^2, the pair of positions is 10 and 12, which correspond to the letters J and L. Thus, JL is the correct answer.

Step-by-Step Solution:

Step 1: Convert letters in CD to positions: C = 3, D = 4. Step 2: Square each position: 3^2 = 9 and 4^2 = 16. Step 3: Place the results side by side: 9 and 16 combine to form 916, which matches the given number. Step 4: For 100144, separate the number into two perfect squares: 100 and 144. Step 5: Take square roots: 100 = 10^2 and 144 = 12^2, so the positions are 10 and 12. Step 6: Convert positions back to letters: 10 → J, 12 → L, giving the pair JL.

Verification / Alternative check:
Check an option like HG. H is the 8th letter and G is the 7th. Squaring these gives 64 and 49, which would form 6449, not 100144. Similarly, MO would give 13^2 = 169 and 15^2 = 225, making 169225. RS corresponds to 18 and 19, giving 324 and 361, which combine to 324361. None of these match 100144. Only JL, with positions 10 and 12, produces 10^2 = 100 and 12^2 = 144, exactly matching the target number.

Why Other Options Are Wrong:
HG, MO, RS, and NP all produce different numerical patterns when the same positional square rule is applied. None of them factor into 100144. The question requires strict adherence to the discovered rule, rather than any loose connection between letters and numbers. Therefore, these options must be rejected.

Common Pitfalls:
A common mistake is to guess based on letter distance or general familiarity instead of fully decoding the number pattern. Another pitfall is to treat 100144 as a single square or cube, which it is not. By carefully splitting the number into two perfect squares and reversing the process back to letter positions, you can reliably find JL as the only pair consistent with the pattern.

Final Answer:
Using the same positional square rule, the pair that matches 100144 is JL, so the completed analogy is “CD : 916 :: JL : 100144”.