Finding the Odd Letter Cluster

The question asks us to identify the letter-cluster that does not follow the same pattern as the others in the given set. This type of problem involves recognizing a rule or relationship between the letters within each cluster, usually based on their positions in the English alphabet.

Let's list the given letter-clusters:

• UCF
• ZXA
• NJM
• HDX

To find the pattern, we will assign numerical values to each letter based on its position in the alphabet (A=1, B=2, ..., Z=26).

Now let's find the positional values for each letter-cluster:

• UCF: U=21, C=3, F=6
• ZXA: Z=26, X=24, A=1
• NJM: N=14, J=10, M=13
• HDX: H=8, D=4, X=24

Analyzing the Pattern in Letter Clusters

We will examine the differences between consecutive letters within each cluster, considering the alphabet to be cyclic (after Z comes A).

1. UCF (21, 3, 6):

• Difference between U and C: From U (21) to C (3). Going forward: U(21) → V(22) → W(23) → X(24) → Y(25) → Z(26) → A(1) → B(2) → C(3). This is a difference of $+8$. Cyclically, this can be calculated as $(3 - 21) \pmod{26} = -18 \pmod{26} = 8$. So, $\text{U} \xrightarrow{+8} \text{C}$.
• Difference between C and F: From C (3) to F (6). C(3) → D(4) → E(5) → F(6). This is a difference of $+3$. $(6 - 3) \pmod{26} = 3$. So, $\text{C} \xrightarrow{+3} \text{F}$.
• Pattern for UCF: $+8, +3$.

2. ZXA (26, 24, 1):

• Difference between Z and X: From Z (26) to X (24). Z(26) → Y(25) → X(24). This is a difference of $-2$. $(24 - 26) \pmod{26} = -2 \pmod{26} = 24$. Going backward is $-2$. So, $\text{Z} \xrightarrow{-2} \text{X}$.
• Difference between X and A: From X (24) to A (1). X(24) → Y(25) → Z(26) → A(1). This is a difference of $+3$. $(1 - 24) \pmod{26} = -23 \pmod{26} = 3$. So, $\text{X} \xrightarrow{+3} \text{A}$.
• Pattern for ZXA: $-2, +3$.

3. NJM (14, 10, 13):

• Difference between N and J: From N (14) to J (10). N(14) → M(13) → L(12) → K(11) → J(10). This is a difference of $-4$. $(10 - 14) \pmod{26} = -4 \pmod{26} = 22$. Going backward is $-4$. So, $\text{N} \xrightarrow{-4} \text{J}$.
• Difference between J and M: From J (10) to M (13). J(10) → K(11) → L(12) → M(13). This is a difference of $+3$. $(13 - 10) \pmod{26} = 3$. So, $\text{J} \xrightarrow{+3} \text{M}$.
• Pattern for NJM: $-4, +3$.

4. HDX (8, 4, 24):

• Difference between H and D: From H (8) to D (4). H(8) → G(7) → F(6) → E(5) → D(4). This is a difference of $-4$. $(4 - 8) \pmod{26} = -4 \pmod{26} = 22$. Going backward is $-4$. So, $\text{H} \xrightarrow{-4} \text{D}$.
• Difference between D and X: From D (4) to X (24). D(4) → C(3) → B(2) → A(1) → Z(26) → Y(25) → X(24). This is a difference of $-4$ (cyclically). $(24 - 4) \pmod{26} = 20$. Going backward is $-4$. So, $\text{D} \xrightarrow{-4} \text{X}$.
• Pattern for HDX: $-4, -4$.

Identifying the Odd Letter Cluster Out

Let's summarize the patterns we found:

• UCF: $+8, +3$
• ZXA: $-2, +3$
• NJM: $-4, +3$
• HDX: $-4, -4$

Upon comparing the patterns, we can see that in UCF, ZXA, and NJM, the difference between the second and third letters is $+3$. In contrast, in HDX, the difference between the second and third letters is $-4$.

Therefore, HDX does not follow the same pattern as the other three letter-clusters.

Revision Table - Letter Cluster Pattern Analysis

Additional Information - Letter Series Reasoning

Letter series and letter cluster problems are common in logical reasoning sections of various competitive exams. They test your ability to identify patterns based on:

• Positional values of letters in the alphabet.
• Differences or sums of positional values.
• Skip patterns (e.g., skipping a fixed number of letters).
• Vowel/Consonant patterns.
• Reversal of alphabet or specific letter groups.

Practicing these types of questions helps improve analytical skills and pattern recognition speed.