Understanding the Letter Triad Logic

This question requires us to identify a specific rule or pattern governing the transformation of letter groups. We are given two examples of these transformations, forming sequences of three letter groups (triads). Our task is to decipher this underlying logic and then select the option that adheres to the same rule.

Analyzing the Provided Examples

Let's first break down the pattern in the given examples:

Example 1: $GAPS$ -> $SPGA$ -> $GPAS$

We can think of the letters in the first word as being in specific positions: G is position 1, A is position 2, P is position 3, and S is position 4.

Step 1: Transformation from $GAPS$ to $SPGA$

Observing the change from $GAPS$ to $SPGA$, we see a rearrangement of letters. If we denote the original positions as $(1, 2, 3, 4)$, the letters in $SPGA$ correspond to the letters originally at positions $(4, 3, 1, 2)$.

• Original: G (1), A (2), P (3), S (4)
• Transformed: S (from 4), P (from 3), G (from 1), A (from 2)

This results in $SPGA$. The positional mapping is: Positions: $(1, 2, 3, 4) \to (4, 3, 1, 2)$

Step 2: Transformation from $SPGA$ to $GPAS$

Now, we consider $SPGA$ as the base, with positions S(1), P(2), G(3), A(4).

The change to $GPAS$ involves another rearrangement. The letters in $GPAS$ correspond to the letters originally at positions $(3, 2, 4, 1)$ within $SPGA$.

• Original: S (1), P (2), G (3), A (4)
• Transformed: G (from 3), P (from 2), A (from 4), S (from 1)

This results in $GPAS$. The positional mapping is: Positions: $(1, 2, 3, 4) \to (3, 2, 4, 1)$

Example 2: $AUNT$ -> $TNAU$ -> $ANUT$

Let's check if this example follows the same two rules:

Step 1: $AUNT$ to $TNAU$

A(1), U(2), N(3), T(4). Applying the first rule (Positions: $(1, 2, 3, 4) \to (4, 3, 1, 2)$) gives T(4), N(3), A(1), U(2), which correctly forms $TNAU$.

Step 2: $TNAU$ to $ANUT$

T(1), N(2), A(3), U(4). Applying the second rule (Positions: $(1, 2, 3, 4) \to (3, 2, 4, 1)$) gives A(3), N(2), U(4), T(1), which correctly forms $ANUT$.

Both examples consistently use the same two-step positional permutation logic.

Evaluating the Options Based on the Pattern

We will now test each provided option against the identified logic:

Option 1: $HAIR$ -> $RIHA$ -> $HIAR$

• Step 1: $HAIR$ to $RIHA$

  H(1), A(2), I(3), R(4). Applying the first rule (Positions: $(1, 2, 3, 4) \to (4, 3, 1, 2)$) yields R(4), I(3), H(1), A(2), resulting in $RIHA$. This matches the option.

• Step 2: $RIHA$ to $HIAR$

  R(1), I(2), H(3), A(4). Applying the second rule (Positions: $(1, 2, 3, 4) \to (3, 2, 4, 1)$) yields H(3), I(2), A(4), R(1), resulting in $HIAR$. This also matches the option.

Since both steps align with the derived logic, Option 1 follows the pattern.

Option 2: $DINE$ -> $DNIE$ -> $ENID$

• Step 1: $DINE$ to $DNIE$

  D(1), I(2), N(3), E(4). Applying the first rule (Positions: $(1, 2, 3, 4) \to (4, 3, 1, 2)$) gives E(4), N(3), D(1), I(2), which forms $ENDI$. The word provided in the option is $DNIE$. This is a mismatch.

As the first transformation doesn't match, Option 2 is incorrect.

Option 3: $DIMS$ -> $IMDS$ -> $IMSD$

• Step 1: $DIMS$ to $IMDS$

  D(1), I(2), M(3), S(4). Applying the first rule (Positions: $(1, 2, 3, 4) \to (4, 3, 1, 2)$) gives S(4), M(3), D(1), I(2), resulting in $SMDI$. The word provided in the option is $IMDS$. This is a mismatch.

As the first transformation doesn't match, Option 3 is incorrect.

Option 4: $HALF$ -> $FLHA$ -> $ALHF$

• Step 1: $HALF$ to $FLHA$

  H(1), A(2), L(3), F(4). Applying the first rule (Positions: $(1, 2, 3, 4) \to (4, 3, 1, 2)$) gives F(4), L(3), H(1), A(2), resulting in $FLHA$. This matches the option.

• Step 2: $FLHA$ to $ALHF$

  F(1), L(2), H(3), A(4). Applying the second rule (Positions: $(1, 2, 3, 4) \to (3, 2, 4, 1)$) gives H(3), L(2), A(4), F(1), resulting in $HLAF$. The word provided in the option is $ALHF$. This is a mismatch.

As the second transformation doesn't match, Option 4 is incorrect.

Final Conclusion

By systematically analyzing the logic demonstrated in the examples (GAPS -> SPGA -> GPAS and AUNT -> TNAU -> ANUT) and applying it to the given choices, we found that only Option 1 (HAIR -> RIHA -> HIAR) perfectly replicates the observed two-step positional permutation pattern.