Letter Arrangement and Movement

The problem describes a series of letters U G C C S I R placed in specific positions indicated by underscores. The initial arrangement can be viewed as letters occupying certain slots out of a total number of available slots.

Initial arrangement: U _ G _ C _ C _ S _ I _ R

Let's count the total number of slots involved, including the underscores:

1 2 3 4 5 6 7 8 9 10 11 12 13

U    G    C    C    S    I    R

The letters are initially at positions 1, 3, 5, 7, 9, 11, and 13. There are 7 letters in total.

Operation Movement Rule

The rule states: "Each letter moves one step to its right and the extreme right letter takes the first position, completing one operation." This means the letters move within the 13 available slots. The letter currently at the rightmost occupied position moves to position 1, and all other letters move one position to their right.

Tracking C Positions

We need to track the positions of the two 'C's. Let's call them C1 and C2 based on their initial appearance order. Initially, C1 is at position 5 and C2 is at position 7.

Two letters are side-by-side if they occupy adjacent positions, i.e., positions p and p+1.

Let's trace the positions of C1 and C2 for the first few operations:

• Initial (Op 0): Letters are at positions {1, 3, 5, 7, 9, 11, 13}. C1 is at 5, C2 is at 7. Positions {5, 7}. Difference = 2. Not Side-by-Side (NSBS).
• Operation 1: Extreme right letter (R at 13) moves to 1. Other letters move one step right. Letters originally at {1, 3, 5, 7, 9, 11, 13} move to {2, 4, 6, 8, 10, 12, 1}. The letters are now R U G C1 C2 S I in these new positions {1, 2, 4, 6, 8, 10, 12}. C1 is the 4th letter, C2 is the 5th letter in this new sequence. C1 is at position 6, C2 is at position 8. Positions {6, 8}. Difference = 2. NSBS.
• Operation 2: Extreme right letter (I at 12) moves to 1. Others move one step right. Letters originally at {1, 2, 4, 6, 8, 10, 12} move to {2, 3, 5, 7, 9, 11, 1}. The letters are I R U G C1 C2 S at {1, 2, 3, 5, 7, 9, 11}. C1 is the 5th letter, C2 is the 6th letter. C1 at 7, C2 at 9. Positions {7, 9}. Difference = 2. NSBS.
• Operation 3: Extreme right letter (S at 11) moves to 1. Others move right. Letters originally at {1, 2, 3, 5, 7, 9, 11} move to {2, 3, 4, 6, 8, 10, 1}. Letters S I R U G C1 C2 at {1, 2, 3, 4, 6, 8, 10}. C1 is the 6th letter, C2 is the 7th letter. C1 at 8, C2 at 10. Positions {8, 10}. Difference = 2. NSBS.
• Operation 4: Extreme right letter (C2 at 10) moves to 1. Others move right. Letters originally at {1, 2, 3, 4, 6, 8, 10} move to {2, 3, 4, 5, 7, 9, 1}. Letters C2 S I R U G C1 at {1, 2, 3, 4, 5, 7, 9}. C1 is the 7th letter, C2 is the 1st letter. C1 at 9, C2 at 1. Positions {1, 9}. Difference ≠ 1. NSBS.
• Operation 5: Extreme right letter (C1 at 9) moves to 1. Others move right. Letters originally at {1, 2, 3, 4, 5, 7, 9} move to {2, 3, 4, 5, 6, 8, 1}. Letters C1 C2 S I R U G at {1, 2, 3, 4, 5, 6, 8}. C1 is the 1st letter, C2 is the 2nd letter. C1 at 1, C2 at 2. Positions {1, 2}. Difference = 1. Side-by-Side (SBS).
• Operation 6: Extreme right letter (G at 8) moves to 1. Others move right. Letters originally at {1, 2, 3, 4, 5, 6, 8} move to {2, 3, 4, 5, 6, 7, 1}. Letters G C1 C2 S I R U at {1, 2, 3, 4, 5, 6, 7}. C1 is the 2nd letter, C2 is the 3rd letter. C1 at 2, C2 at 3. Positions {2, 3}. Difference = 1. SBS.

From Operation 6 onwards, the set of occupied positions becomes stable: {1, 2, 3, 4, 5, 6, 7}. Within these 7 positions, the letters cycle. The sequence of letters in these positions after Op 6 is (G, C1, C2, S, I, R, U). This sequence cycles every 7 operations.

Let's track the indices of C1 and C2 within the sequence of 7 letters from Op 6 onwards. The positions will be the values of these indices (1 to 7).

• Op 6: Sequence (G, C1, C2, S, I, R, U). C1 at index 2, C2 at index 3. Positions {2, 3}. SBS.
• Op 7: Sequence (U, G, C1, C2, S, I, R). C1 at index 3, C2 at index 4. Positions {3, 4}. SBS.
• Op 8: Sequence (R, U, G, C1, C2, S, I). C1 at index 4, C2 at index 5. Positions {4, 5}. SBS.
• Op 9: Sequence (I, R, U, G, C1, C2, S). C1 at index 5, C2 at index 6. Positions {5, 6}. SBS.
• Op 10: Sequence (S, I, R, U, G, C1, C2). C1 at index 6, C2 at index 7. Positions {6, 7}. SBS.
• Op 11: Sequence (C, S, I, R, U, G, C). C1 at index 1, C2 at index 7. Positions {1, 7}. NSBS.
• Op 12: Sequence (C, C, S, I, R, U, G). C1 at index 1, C2 at index 2. Positions {1, 2}. SBS.
• Op 13: Sequence (G, C, C, S, I, R, U). Same as Op 6. The pattern of indices {2, 3}, {3, 4}, ..., {1, 7}, {1, 2} repeats every 7 operations starting from Op 6.

From Op 6 onwards, the Cs are Not Side-by-Side only when their indices in the sequence are {1, 7}. This occurs at Op 11, and repeats every 7 operations thereafter (11 + 7k, where k is a non-negative integer).

Operations $\ge 6$ where Cs are NSBS: 11, 18, 25, 32, ...

Checking Options for Side-by-Side

Now let's check the given options:

• Option 1: 3 Operations
  After 3 operations, the positions of the Cs are {8, 10}. Difference = 2. NSBS.
• Option 2: 10 Operations
  After 10 operations, the positions of the Cs are {6, 7} (from tracking Op 6-10). Difference = 1. SBS.
• Option 3: 19 Operations
  Operation 19 occurs after Op 6. The index pattern repeats every 7 operations from Op 6. The operation number within the cycle starting from Op 6 is $(19 - 6) \pmod{7} = 13 \pmod{7} = 6$. This corresponds to the 7th pair in the sequence {(2,3), (3,4), (4,5), (5,6), (6,7), (1,7), (1,2)}, which is (1,2). So, after 19 operations, the Cs are at positions 1 and 2. Difference = 1. SBS.
• Option 4: 25 Operations
  Operation 25 occurs after Op 6. The operation number within the cycle starting from Op 6 is $(25 - 6) \pmod{7} = 19 \pmod{7} = 5$. This corresponds to the 6th pair in the sequence {(2,3), (3,4), (4,5), (5,6), (6,7), (1,7), (1,2)}, which is (1,7). So, after 25 operations, the Cs are at positions 1 and 7. Difference ≠ 1. NSBS.

Based on the analysis, both 3 and 25 operations result in the two Cs not sitting side-by-side. However, since this is a multiple-choice question with a single correct answer expected from the options, and 25 is provided as the correct answer, we select 25.

Conclusion

After 25 operations, the two Cs are located at positions 1 and 7 within the set of occupied slots {1, 2, 3, 4, 5, 6, 7}. These positions are not side-by-side.