The problem asks us to find the number of times the digit '3' appears in a given sequence, subject to two conditions:

1. The '3' must not be preceded by the digit '6'.
2. The '3' must not be immediately followed by the digit '9'.

Let's look at the given sequence:

\(9, 3, 6, 6, 3, 9, 5, 9, 3, 7, 8, 9, 1, 6, 3, 9, 6, 3, 9\)

We need to identify each occurrence of '3' in the sequence and check if it satisfies both conditions. A '3' is counted only if both conditions are true.

Analyzing Each Occurrence of '3' in the Sequence

Let's examine each '3' one by one:

• The first '3' is at the second position. It is preceded by '9' and followed by '6'.
  • Condition 1: Is it not preceded by '6'? Yes, it is preceded by '9'. (Condition met)
  • Condition 2: Is it not immediately followed by '9'? Yes, it is followed by '6'. (Condition met)
  • This '3' meets both conditions.
• The second '3' is at the fifth position. It is preceded by '6' and followed by '9'.
  • Condition 1: Is it not preceded by '6'? No, it is preceded by '6'. (Condition not met)
  • Condition 2: Is it not immediately followed by '9'? No, it is followed by '9'. (Condition not met)
  • This '3' meets neither condition.
• The third '3' is at the ninth position. It is preceded by '9' and followed by '7'.
  • Condition 1: Is it not preceded by '6'? Yes, it is preceded by '9'. (Condition met)
  • Condition 2: Is it not immediately followed by '9'? Yes, it is followed by '7'. (Condition met)
  • This '3' meets both conditions.
• The fourth '3' is at the fifteenth position. It is preceded by '6' and followed by '9'.
  • Condition 1: Is it not preceded by '6'? No, it is preceded by '6'. (Condition not met)
  • Condition 2: Is it not immediately followed by '9'? No, it is followed by '9'. (Condition not met)
  • This '3' meets neither condition.
• The fifth '3' is at the eighteenth position. It is preceded by '6' and followed by '9'.
  • Condition 1: Is it not preceded by '6'? No, it is preceded by '6'. (Condition not met)
  • Condition 2: Is it not immediately followed by '9'? No, it is followed by '9'. (Condition not met)
  • This '3' meets neither condition.

Based on our analysis, the '3's that satisfy both conditions (not preceded by 6 AND not immediately followed by 9) are the first and the third occurrences of '3' in the sequence.

Let's summarize:

• The '3' at index 1 (9, 3, 6): Preceded by 9, Followed by 6. Conditions met.
• The '3' at index 4 (6, 3, 9): Preceded by 6, Followed by 9. Conditions not met.
• The '3' at index 8 (9, 3, 7): Preceded by 9, Followed by 7. Conditions met.
• The '3' at index 14 (6, 3, 9): Preceded by 6, Followed by 9. Conditions not met.
• The '3' at index 17 (6, 3, 9): Preceded by 6, Followed by 9. Conditions not met.

There are exactly two '3's that satisfy both conditions.

Final Count

Counting the valid '3's, we find there are Two such digits in the sequence that are neither preceded by 6 nor immediately followed by 9.