In the following letter analogy, AEDM is related to ZQRN. Using the same alphabetical pattern, FLMO is related to which of the following letter groups?

Introduction / Context:
This is a non trivial letter analogy that focuses on positions of letters in the English alphabet and how pairs of letters can be related through sums of their positions. The first pair AEDM and ZQRN encodes a pattern that is not a simple uniform shift, so the learner needs to think in terms of complementary positions. Once this rule is discovered, the same pattern must be applied to the second group FLMO to obtain the missing term from the alternatives.

Given Data / Assumptions:

• First pair of letter groups: AEDM : ZQRN
• Second first term: FLMO
• We must choose the group that stands in the same relation to FLMO as ZQRN stands to AEDM.
• We use A to Z mapped to positions 1 to 26.

Concept / Approach:
When a constant forward or backward shift does not explain the mapping between the two groups, another common idea is to look at sums of letter positions that equal a fixed total. For AEDM to ZQRN, if we pair letters position wise and add their positions, we notice that the sums follow a pattern. The first and fourth positions sum to 27, while the second and third positions sum to 22. This alternating sum rule is the key to unlocking the analogy.

Step-by-Step Solution:
Step 1: Convert AEDM into positions. A is 1, E is 5, D is 4, and M is 13. Step 2: Convert ZQRN into positions. Z is 26, Q is 17, R is 18, and N is 14. Step 3: Add positions in each column. For the first letters, 1 + 26 = 27. For the second letters, 5 + 17 = 22. For the third letters, 4 + 18 = 22. For the fourth letters, 13 + 14 = 27. So the mapping is defined by sums of 27, 22, 22, and 27 at positions 1, 2, 3, and 4 respectively. Step 4: Apply the same sum pattern to FLMO. Convert FLMO to positions: F is 6, L is 12, M is 13, O is 15. Step 5: Find missing letters so that first and fourth positions sum to 27 and second and third positions sum to 22. For the first position, 6 + x = 27 gives x = 21, which is U. For the second position, 12 + x = 22 gives x = 10, which is J. For the third position, 13 + x = 22 gives x = 9, which is I. For the fourth position, 15 + x = 27 gives x = 12, which is L. Step 6: Combine these letters to get the related group UJIL.

Verification / Alternative check:
We can quickly verify that our constructed group UJIL indeed follows the same complement rule. For the new pair FLMO and UJIL the sums by position are: F (6) plus U (21) equals 27, L (12) plus J (10) equals 22, M (13) plus I (9) equals 22, and O (15) plus L (12) equals 27. This exactly mirrors the pattern seen in AEDM and ZQRN. None of the other options produce the required pair of sums and therefore they do not match the analogy structure.

Why Other Options Are Wrong:
Option UQIL fails because L plus Q does not equal 22 and some position sums break the required values of 27 or 22. Option VJIM also gives incorrect sums when positions are added with FLMO. Option UJLM changes the order of letters so that the second and third position sums are no longer 22, which breaks the pattern. The question demands an exact replication of the sum pattern 27, 22, 22, 27, and only UJIL satisfies that requirement.

Common Pitfalls:
Many learners initially try simple constant shifts such as plus two or minus three and give up when these do not work. For harder letter analogies it is important to consider alternative relationships such as complementary positions adding to a fixed constant, symmetric positions around the middle of the alphabet, or patterns in the differences between letters. Another pitfall is checking only one or two positions instead of verifying the rule for all positions in the group. A robust solution always confirms the pattern column by column.

Final Answer:
Using the same complement pattern as in AEDM : ZQRN, the group of letters related to FLMO is UJIL.