Series Digit Pattern

We need to analyze the given number series to find specific digit patterns. The series is:

\(3 5 6 3 4 6 4 3 1 5 7 6 8 6 3 1 3 6 7 8 9 7 5\)

The criteria for counting are:

• The digit must be odd (1, 3, 5, 7, 9).
• It must be immediately preceded by an even digit (2, 4, 6, 8).
• It must be immediately followed by an odd digit (1, 3, 5, 7, 9).

This translates to finding the pattern: Even - Odd - Odd.

Odd Digit Identification

Let's scan the series and identify triplets that match the Even - Odd - Odd pattern:

• \(... 4 3 1 ...\) : The central digit '3' is odd. It's preceded by '4' (even) and followed by '1' (odd). This is a match.
• \(... 6 3 1 ...\) : The central digit '3' is odd. It's preceded by '6' (even) and followed by '1' (odd). This is a match.
• \(... 8 9 7 ...\) : The central digit '9' is odd. It's preceded by '8' (even) and followed by '7' (odd). This is a match.

Checking other potential triplets confirms no other matches:

• \(6 3 4\) is Even-Odd-Even.
• \(4 6 4\) is Even-Even-Even.
• \(6 4 3\) is Even-Even-Odd (middle is even).
• \(3 1 5\) is Odd-Odd-Odd (preceding is odd).
• \(7 6 8\) is Odd-Even-Even.
• \(8 6 3\) is Even-Even-Odd (middle is even).
• \(3 6 7\) is Odd-Even-Odd (preceding is odd).
• \(6 7 8\) is Even-Odd-Even.
• \(7 8 9\) is Odd-Even-Odd (preceding is odd).

Count Calculation

We identified exactly three instances where an odd digit is immediately preceded by an even digit and immediately followed by an odd digit.

The qualifying digits are:

• The first '3' (in 4 3 1)
• The second '3' (in 6 3 1)
• The '9' (in 8 9 7)

Thus, the total count is THREE.