This question describes a symbol arrangement mechanism where symbols are rearranged step-by-step following specific rules. An example sequence shows how the rearrangement happens across multiple steps. We are given the arrangement at the first step for a new set of symbols and need to find at which subsequent step a target arrangement appears.

Understanding the Symbol Arrangement Rules

First, let's carefully analyze the provided example sequence to understand the rules applied at each step. The example is:

Input → N, T, Y, A, P, F

Step 1 → Y, T, F, A, P, N

Step 2 → Y, P, F, T, A, N

Step 3 → P, F, A, T, Y, N

Step 4 → P, T, N, F, Y, A

Step 5 → P, N, T, A, Y, F

We need to determine the rule that transforms the arrangement from one step to the next.

• Rule 1 (Input to Step 1): Comparing Input (N, T, Y, A, P, F) and Step 1 (Y, T, F, A, P, N):
  • Symbol at position 3 (Y) moves to position 1.
  • Symbol at position 6 (F) moves to position 3.
  • Symbol at position 1 (N) moves to position 6.
  • Symbols at positions 2 (T), 4 (A), and 5 (P) remain in their positions.
  • This is a cyclic movement for positions (1, 3, 6): 1 → 6, 3 → 1, 6 → 3.
• Rule 2 (Step 1 to Step 2): Comparing Step 1 (Y, T, F, A, P, N) and Step 2 (Y, P, F, T, A, N):
  • Symbol at position 1 (Y) stays at position 1.
  • Symbol at position 2 (T) moves to position 4.
  • Symbol at position 3 (F) stays at position 3.
  • Symbol at position 4 (A) moves to position 5.
  • Symbol at position 5 (P) moves to position 2.
  • Symbol at position 6 (N) stays at position 6.
  • This is a cyclic movement for positions (2, 4, 5): 2 → 4, 4 → 5, 5 → 2. Positions 1, 3, 6 stay.
• Rule 3 (Step 2 to Step 3): Comparing Step 2 (Y, P, F, T, A, N) and Step 3 (P, F, A, T, Y, N):
  • Symbol at position 1 (Y) moves to position 5.
  • Symbol at position 2 (P) moves to position 1.
  • Symbol at position 3 (F) moves to position 2.
  • Symbol at position 4 (T) stays at position 4.
  • Symbol at position 5 (A) moves to position 3.
  • Symbol at position 6 (N) stays at position 6.
  • This is a cyclic movement for positions (1, 2, 3, 5): 1 → 5, 2 → 1, 3 → 2, 5 → 3. Positions 4, 6 stay.
• Rule 4 (Step 3 to Step 4): Comparing Step 3 (P, F, A, T, Y, N) and Step 4 (P, T, N, F, Y, A):
  • Symbol at position 1 (P) stays at position 1.
  • Symbol at position 2 (F) moves to position 4.
  • Symbol at position 3 (A) moves to position 6.
  • Symbol at position 4 (T) moves to position 2.
  • Symbol at position 5 (Y) stays at position 5.
  • Symbol at position 6 (N) moves to position 3.
  • This is a cyclic movement for positions (2, 3, 4, 6): 2 → 4, 3 → 6, 4 → 2, 6 → 3. Positions 1, 5 stay.
• Rule 5 (Step 4 to Step 5): Comparing Step 4 (P, T, N, F, Y, A) and Step 5 (P, N, T, A, Y, F):
  • Symbol at position 1 (P) stays at position 1.
  • Symbol at position 2 (T) moves to position 3.
  • Symbol at position 3 (N) moves to position 2.
  • Symbol at position 4 (F) moves to position 6.
  • Symbol at position 5 (Y) stays at position 5.
  • Symbol at position 6 (A) moves to position 4.
  • This involves swaps: (Position 2 ↔ Position 3) and (Position 4 ↔ Position 6). Positions 1, 5 stay.

Applying the Symbol Arrangement Rules

We are given that the arrangement at the first step of some input is O, D, K, W, G, X. We need to apply the rules derived above sequentially to find the subsequent steps until we reach the target arrangement G, X, D, W, O, K.

Starting arrangement:

Step 1: O, D, K, W, G, X

Apply Rule 2 (2→4, 4→5, 5→2; 1,3,6 stay) to get Step 2:

Step 2: O, G, K, D, W, X

Apply Rule 3 (1→5, 2→1, 3→2, 5→3; 4,6 stay) to get Step 3:

Step 3: G, K, W, D, O, X

Apply Rule 4 (2→4, 3→6, 4→2, 6→3; 1,5 stay) to get Step 4:

Step 4: G, D, X, K, O, W

Apply Rule 5 (2↔3 swap, 4↔6 swap; 1,5 stay) to get Step 5:

Step 5: G, X, D, W, O, K

Conclusion

Comparing the arrangement at each step with the target arrangement G, X, D, W, O, K:

• Step 1: O, D, K, W, G, X (Does not match)
• Step 2: O, G, K, D, W, X (Does not match)
• Step 3: G, K, W, D, O, X (Does not match)
• Step 4: G, D, X, K, O, W (Does not match)
• Step 5: G, X, D, W, O, K (Matches the target arrangement)

The target arrangement G, X, D, W, O, K is displayed at the 5^th step.

The final answer is “5^th”.