In this letter analogy, “HMPU is to IOSY as GMRF is to ______”. Choose the group of letters that results from applying the same increasing alphabet shift pattern to GMRF.

Introduction / Context:
This is a coded letter analogy where each position in a group of letters is shifted forward by a specific amount. The example pair “HMPU : IOSY” reveals a pattern in how each letter moves through the alphabet. We must uncover that pattern and then apply it to the second group “GMRF” to find the correct related group from the given options. Questions like this test careful observation of positional changes in the alphabet.

Given Data / Assumptions:

• Example pair: HMPU maps to IOSY. • Target mapping: GMRF maps to ? • Options: FKOB, HOUJ, HPUJ, HOJU. • We assume the English alphabet with A = 1, B = 2, ..., Z = 26 and simple forward shifts.

Concept / Approach:
We look at each corresponding letter in HMPU and IOSY and compute how many positions forward it moves in the alphabet. If the pattern is consistent across positions, we can treat it as the rule of transformation. In this case, the shifts increase by one at each step: the first letter moves by +1, the second by +2, the third by +3, and the fourth by +4. Once this pattern is confirmed, we apply the same set of shifts to the letters in GMRF and then match the result against the answer choices.

Step-by-Step Solution:
Step 1: Convert HMPU and IOSY to alphabet positions. H = 8, M = 13, P = 16, U = 21. I = 9, O = 15, S = 19, Y = 25. Step 2: Compute the shifts. 8 → 9: +1, 13 → 15: +2, 16 → 19: +3, 21 → 25: +4. So the pattern is “+1, +2, +3, +4”. Step 3: Apply the same pattern to GMRF. G = 7, M = 13, R = 18, F = 6. Add shifts: 7 + 1 = 8 → H, 13 + 2 = 15 → O, 18 + 3 = 21 → U, 6 + 4 = 10 → J. Step 4: The resulting sequence is H O U J (HOUJ).

Verification / Alternative check:
To verify, you can perform the reverse operation on IOSY. Subtract 1 from I to get H, subtract 2 from O to get M, subtract 3 from S to get P, and subtract 4 from Y to get U. This confirms the transformation is consistent. Doing the same reverse shifts on HOUJ returns GMRF, which shows that we have applied the rule correctly in both directions. Among the answer choices, only HOUJ matches the output of this precise set of shifts.

Why Other Options Are Wrong:
• FKOB, HPUJ, HOJU: These groups contain some of the same letters but in different orders or with incorrect letters, and they do not result from applying the exact +1, +2, +3, +4 shift pattern to GMRF. For a valid analogy, every position must follow the same transformation rule as in the example pair.

Common Pitfalls:
Many learners check only the first or second letter and pick an option that partly matches, without verifying all positions. Others assume a constant shift like +2 without noticing the progressive nature of the pattern. Always compute shifts for each letter carefully and confirm that the same pattern explains the entire example pair before using it on the new group.

Final Answer:
The letter group that correctly completes the analogy is HOUJ.