Four letter-clusters have been given, out of which three are alike in some manner and one is different. Select the letter-cluster that is different.
NRJG
Pattern Analysis of Letter Clusters
The question asks us to identify the letter cluster that is different from the others among the given options. To solve this, we first assign numerical positions to each letter of the alphabet (A=1, B=2, ..., Z=26). Then, we analyze the pattern of differences between consecutive letters within each cluster.
Letter Position Mapping
We map each letter to its corresponding position in the alphabet:
- A=1, B=2, C=3, D=4, E=5, F=6, G=7, H=8, I=9, J=10, K=11, L=12, M=13, N=14, O=15, P=16, Q=17, R=18, S=19, T=20, U=21, V=22, W=23, X=24, Y=25, Z=26.
Calculating Differences Between Consecutive Letters
Let's calculate the difference in alphabetical positions between consecutive letters for each cluster:
| Cluster | Step 1: Letter 1 → Letter 2 | Step 2: Letter 2 → Letter 3 | Step 3: Letter 3 → Letter 4 |
|---|---|---|---|
| NRJG | $N_{14} \rightarrow R_{18} = +4$ | $R_{18} \rightarrow J_{10} = -8$ | $J_{10} \rightarrow G_{7} = -3$ |
| DHSP | $D_{4} \rightarrow H_{8} = +4$ | $H_{8} \rightarrow S_{19} = +11$ | $S_{19} \rightarrow P_{16} = -3$ |
| FJQN | $F_{6} \rightarrow J_{10} = +4$ | $J_{10} \rightarrow Q_{17} = +7$ | $Q_{17} \rightarrow N_{14} = -3$ |
| TXCZ | $T_{20} \rightarrow X_{24} = +4$ | $X_{24} \rightarrow C_{3} = +5 \pmod{26}$ | $C_{3} \rightarrow Z_{26} = +23$ |
Note: For TXCZ, the difference from X (24) to C (3) is calculated modulo 26. The shortest path is $X \rightarrow Y \rightarrow Z \rightarrow A \rightarrow B \rightarrow C$, which corresponds to a difference of +5.
Identifying the Odd Letter Cluster
By comparing the sequences of differences obtained:
- NRJG: (+4, -8, -3)
- DHSP: (+4, +11, -3)
- FJQN: (+4, +7, -3)
- TXCZ: (+4, +5, +23)
We can observe a pattern related to the difference between the second and third letters in each cluster:
- In NRJG, the difference $R \rightarrow J$ is -8 (negative).
- In DHSP, the difference $H \rightarrow S$ is +11 (positive).
- In FJQN, the difference $J \rightarrow Q$ is +7 (positive).
- In TXCZ, the difference $X \rightarrow C$ is +5 (positive, considering wrap-around).
The cluster NRJG is different because the step between the second and third letters results in a negative difference (-8), whereas the other clusters (DHSP, FJQN, TXCZ) exhibit a positive difference for this step.
Conclusion
Based on the analysis of the differences between consecutive letter positions, the letter cluster NRJG stands out as different from the rest due to the negative difference calculated between its second and third letters.
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