This question asks us to identify the relationship between letter clusters and apply that same relationship to find a missing cluster. This type of problem falls under the category of letter-cluster analogies or coding-decoding in reasoning.

Understanding the Letter Cluster Analogy

In a letter cluster analogy, the letters in the first cluster are related to the letters in the second cluster based on a specific rule or pattern. This pattern could involve shifting letter positions in the alphabet, skipping letters, or other logical operations. The same pattern must apply to the third and fourth clusters, and we need to use it to connect the fifth cluster to the missing sixth cluster.

Finding the Pattern in the First Pair: FUELED : DXBOAG

Let's examine the transformation from FUELED to DXBOAG by looking at the alphabetical position of each letter and the difference in positions.

The shifts observed are: $-2, +3, -3, +3, -4, +3$. It appears there's an alternating pattern: a decreasing negative shift followed by a constant positive shift of $+3$.

Verifying the Pattern with the Second Pair: GELATO : EHIDPR

Let's check if the same pattern holds true for the second pair, GELATO and EHIDPR.

The shifts for the second pair are also: $-2, +3, -3, +3, -4, +3$. The pattern is confirmed.

Applying the Pattern to the Third Pair: HEARTY : ?

Now, we apply the established pattern of shifts ($-2, +3, -3, +3, -4, +3$) to the letters in the cluster HEARTY.

Applying the shifts results in the letters F, H, X, U, P, B.

Combining these letters gives the cluster FHXUPB.

Conclusion

Based on the consistent pattern of letter shifts ($-2, +3, -3, +3, -4, +3$), the cluster related to HEARTY is FHXUPB.

Revision Table: Letter Shifts Pattern

Here is a summary of the pattern applied to each letter position in the cluster:

• 1st Letter: Shift by $-2$
• 2nd Letter: Shift by $+3$
• 3rd Letter: Shift by $-3$
• 4th Letter: Shift by $+3$
• 5th Letter: Shift by $-4$
• 6th Letter: Shift by $+3$

Additional Information: Letter Reasoning Concepts

Letter reasoning questions like this one test your ability to identify logical patterns. Common patterns include:

• Positional Shifts: Adding or subtracting a fixed number or a varying sequence of numbers to the letter's alphabetical position.
• Alphabetical Series: Patterns based on sequences like A, C, E (skipping one letter) or A, B, D, G (adding increasing differences).
• Reverse Alphabetical Order: Using the alphabet backward (Z, Y, X...).
• Vowel/Consonant Patterns: Rules applying specifically to vowels or consonants.
• Letter Reversal: Reversing the order of letters within a group or the entire word/cluster.
• Combination of Patterns: Sometimes multiple rules are combined.

Practicing with different types of letter reasoning questions helps improve pattern recognition skills.