Identifying the Odd Letter-Cluster Pair

The task is to identify the letter-cluster pair that differs from the others based on a specific alphabetical relationship. The pattern is not related to vowels, consonants, or simple letter positions.

Letter-to-Number Mapping

We assign numerical values to letters based on their order in the English alphabet (A=1, Z=26).

Pattern: Cyclic Alphabetical Distance

The grouping is based on the cyclic distance between the letters of the first cluster and the letters of the second cluster.

For any two letters $L_1$ and $L_2$ with positions $P_1$ and $P_2$, the cyclic distance is calculated as $\min(|P_1 - P_2|, 26 - |P_1 - P_2|)$.

The common pattern identified is that the cyclic distance between the first letters of the pairs is 5, and the cyclic distance between the second letters of the pairs is 2.

Analysis of Each Pair

1. TM - OO

Letters: T(20), M(13) and O(15), O(15).

• Distance (T, O): $\min(|20 - 15|, 26 - |20 - 15|) = \min(5, 21) = 5$.
• Distance (M, O): $\min(|13 - 15|, 26 - |13 - 15|) = \min(2, 24) = 2$.

This pair follows the (5, 2) pattern.

2. CP - XS

Letters: C(3), P(16) and X(24), S(19).

• Distance (C, X): $\min(|3 - 24|, 26 - |3 - 24|) = \min(21, 5) = 5$.
• Distance (P, S): $\min(|16 - 19|, 26 - |16 - 19|) = \min(3, 23) = 3$.

This pair follows the (5, 3) pattern.

3. NH - IJ

Letters: N(14), H(8) and I(9), J(10).

• Distance (N, I): $\min(|14 - 9|, 26 - |14 - 9|) = \min(5, 21) = 5$.
• Distance (H, J): $\min(|8 - 10|, 26 - |8 - 10|) = \min(2, 24) = 2$.

This pair follows the (5, 2) pattern.

4. IY - DA

Letters: I(9), Y(25) and D(4), A(1).

• Distance (I, D): $\min(|9 - 4|, 26 - |9 - 4|) = \min(5, 21) = 5$.
• Distance (Y, A): $\min(|25 - 1|, 26 - |25 - 1|) = \min(24, 2) = 2$.

This pair follows the (5, 2) pattern.

Result

Three pairs (TM-OO, NH-IJ, IY-DA) follow the (5, 2) cyclic distance pattern. The pair CP-XS follows a (5, 3) pattern and is therefore the odd one out.