In the following letter analogy, XY is related to AB in a particular cyclic way around the alphabet. Using the same pattern, which pair of letters should complete the analogy XY : AB :: LM : ?

Introduction / Context:
This analogy uses cyclic movement around the alphabet where letters after Z wrap around to the beginning again. In the pair XY : AB, each letter in XY is shifted forward by a fixed number of positions to reach the corresponding letter in AB. We must identify that shift and then apply it to LM to find the correct pair that completes the analogy. This tests your ability to work with modular arithmetic on the alphabet circle.

Given Data / Assumptions:
First pair: XY : AB. Second pair: LM : ?. All letters are capital letters from A to Z. After Z, counting continues from A again, forming a cycle.

Concept / Approach:
We convert X, Y, A, and B into their numerical positions and calculate the forward shift needed to move from the first pair to the second. Because A and B occur after X and Y when counting cyclically, we must use wrap around logic. Once we determine that the shift is +3 positions, we apply the same +3 shift to L and M. The crucial idea is to treat the alphabet as circular rather than stopping at Z.

Step-by-Step Solution:
Step 1: Determine positions for X, Y, A, and B. X = 24, Y = 25. A = 1, B = 2. Step 2: Find the effective forward shift using wrap around. From X (24) to A (1): moving forward, we pass through Y (25), Z (26), and then A (1). Counting these steps: 24 to 25 is 1, 25 to 26 is 1, 26 to 1 is 1, for a total of 3 steps. From Y (25) to B (2): 25 to 26 is 1, 26 to 1 is 1, 1 to 2 is 1, again total 3 steps. So the rule is: move each letter forward by 3 positions cyclically. Step 3: Apply the same +3 shift to LM. L = 12, plus 3 gives 15, which is O. M = 13, plus 3 gives 16, which is P. Therefore, LM becomes OP.

Verification / Alternative check:
To verify, we can check the reverse process. If we move O and P backward by 3 positions, we should return to L and M. O (15) minus 3 gives 12, which is L. P (16) minus 3 gives 13, which is M. This exact reversal mirrors how A and B move back to X and Y when we subtract 3 with wrap around. Thus, OP is consistent with the cyclic +3 pattern demonstrated by XY and AB.

Why Other Options Are Wrong:
PO reverses the order of the letters and does not come from adding 3 to both L and M. RS and TU correspond to larger shifts from LM and cannot be produced by a uniform +3 step. NO would result from a +2 shift on L and M, which does not match the +3 used in the XY to AB transformation.

Common Pitfalls:
Some students forget to handle the wrap around at Z and mistakenly treat the alphabet as ending there, which can make shifts from X and Y to A and B seem confusing. Another pitfall is to only check the shift for one letter and not verify it for the others. In cyclic alphabet questions, always count steps carefully across Z and ensure that the same number of steps applies position wise to every letter before applying the rule to a new pair.

Final Answer:
Using the same cyclic +3 shift, LM should be related to OP.