Identifying the Odd Letter Cluster Pair

The question asks us to find the letter-cluster pair that does not follow the same logical pattern as the other three, based on English alphabetical order. The note clarifies that the pattern is not related to vowels/consonants or their positions.

Analyzing the Alphabetical Pattern

Let's determine the pattern by analyzing the relationship between the letters within each cluster and between the two clusters. We'll use the standard alphabetical positions (A=1, B=2, ... Z=26).

Step 1: Analyze each pair

We examine the interval between letters within each cluster and the interval between corresponding letters of the two clusters.

• DG - IJ
  • Cluster 1 (DG): D (4), G (7). Interval = $7 - 4 = 3$.
  • Cluster 2 (IJ): I (9), J (10). Interval = $10 - 9 = 1$.
  • Between Clusters: I (9) is $9 - 4 = 5$ positions after D (4). J (10) is $10 - 7 = 3$ positions after G (7).
• HK - LO
  • Cluster 1 (HK): H (8), K (11). Interval = $11 - 8 = 3$.
  • Cluster 2 (LO): L (12), O (15). Interval = $15 - 12 = 3$.
  • Between Clusters: L (12) is $12 - 8 = 4$ positions after H (8). O (15) is $15 - 11 = 4$ positions after K (11).
• PS - TW
  • Cluster 1 (PS): P (16), S (19). Interval = $19 - 16 = 3$.
  • Cluster 2 (TW): T (20), W (23). Interval = $23 - 20 = 3$.
  • Between Clusters: T (20) is $20 - 16 = 4$ positions after P (16). W (23) is $23 - 19 = 4$ positions after S (19).
• NQ - RU
  • Cluster 1 (NQ): N (14), Q (17). Interval = $17 - 14 = 3$.
  • Cluster 2 (RU): R (18), U (21). Interval = $21 - 18 = 3$.
  • Between Clusters: R (18) is $18 - 14 = 4$ positions after N (14). U (21) is $21 - 17 = 4$ positions after Q (17).

Step 2: Identify the Consistent Pattern

The pairs HK - LO, PS - TW, and NQ - RU share a consistent pattern:

• The interval between the two letters within each cluster is 3 ($+3$).
• The first letter of the second cluster is 4 positions after the first letter of the first cluster ($+4$).
• The second letter of the second cluster is 4 positions after the second letter of the first cluster ($+4$).

Mathematically, if the first cluster is $X$ and $X+3$, the second cluster is $Y$ and $Y+3$, where $Y = X+4$. This means $(X+4)$ and $(X+3)+4$ are the letters in the second cluster.

Step 3: Find the Exception

The pair DG - IJ breaks this pattern:

• The interval within the first cluster (DG) is $3$ (D+3=G), which fits.
• However, the interval within the second cluster (IJ) is only $1$ ($I+1=J$), not $3$.
• Additionally, the interval between the first letters (D to I) is $5$, not $4$, and the interval between the second letters (G to J) is $3$, not $4$.

Since DG - IJ does not follow the established $+3$ interval within clusters and $+4$ interval between corresponding cluster letters, it is the odd one out.