Understanding Letter Cluster Analogies

This problem presents a letter analogy. We need to figure out the relationship between the first pair of letter clusters (TAKE : GZPV) and the second pair (NOTE : MLGV). Once we understand this relationship, we apply it to the third pair (FAST : ?) to find the correct answer.

Analyzing the Pattern: TAKE to GZPV

Let's map the letters to their corresponding positions in the alphabet (A=1, B=2, ..., Z=26). We'll then compare the positions in 'TAKE' with those in 'GZPV'.

Observing the positions, we can see a pattern related to the reverse alphabet. In the English alphabet, if you pair letters starting from the beginning and end (A-Z, B-Y, C-X, etc.), the sum of their positions is always 27. For example, A (1) + Z (26) = 27, B (2) + Y (25) = 27.

Let's check if this holds true:

• For T (20): The reverse letter's position should be $27 - 20 = 7$. The 7th letter is G. This matches.
• For A (1): The reverse letter's position should be $27 - 1 = 26$. The 26th letter is Z. This matches.
• For K (11): The reverse letter's position should be $27 - 11 = 16$. The 16th letter is P. This matches.
• For E (5): The reverse letter's position should be $27 - 5 = 22$. The 22nd letter is V. This matches.

So, the rule is to replace each letter with its reverse alphabet counterpart.

Verifying the Pattern: NOTE to MLGV

Let's confirm this reverse alphabet rule using the second pair, NOTE to MLGV.

• N (14): Reverse position is $27 - 14 = 13$. The 13th letter is M.
• O (15): Reverse position is $27 - 15 = 12$. The 12th letter is L.
• T (20): Reverse position is $27 - 20 = 7$. The 7th letter is G.
• E (5): Reverse position is $27 - 5 = 22$. The 22nd letter is V.

The transformation NOTE $\rightarrow$ MLGV perfectly follows the reverse alphabet rule.

Applying the Pattern to FAST

Now we apply the same reverse alphabet transformation rule to the word FAST.

Following the pattern, FAST transforms into UZHG.

Final Answer Derivation

The letter cluster related to FAST, following the reverse alphabet cipher established by the other pairs, is UZHG.