Analyze Number Series for Specific Odd Digit Patterns

This solution explains how to identify specific occurrences of digits within a given number series based on certain conditions. We will analyze the provided series step-by-step to find digits that meet the required criteria.

Understanding the Counting Rule

The question asks us to count the number of odd digits that satisfy two conditions simultaneously:

• The digit must be immediately preceded by an odd digit.
• The digit must be immediately followed by an odd digit.

In simpler terms, we are looking for the pattern Odd Digit - Odd Digit - Odd Digit within the series. The digit we count is the middle one in this three-digit sequence.

Step-by-Step Series Examination

The given number series is:

$4 7 5 8 9 2 8 7 1 6 8 2 9 8 5 2 8 2 6 4 4 7 8$

Let's identify the parity (Even or Odd) of each digit in the series:

$4(E) 7(O) 5(O) 8(E) 9(O) 2(E) 8(E) 7(O) 1(O) 6(E) 8(E) 2(E) 9(O) 8(E) 5(O) 2(E) 8(E) 2(E) 6(E) 4(E) 4(E) 7(O) 8(E)$

Now, we examine every possible group of three consecutive digits (Preceding, Current, Following) to see if the Current Digit is Odd, and if it is preceded by an Odd digit and followed by an Odd digit.

Checking Consecutive Triplets

• 4 7 5: Current digit is 7 (Odd). Preceded by 4 (Even). Fails because the preceding digit is Even.
• 7 5 8: Current digit is 5 (Odd). Preceded by 7 (Odd). Followed by 8 (Even). Fails because the following digit is Even.
• 5 8 9: The current digit is 8 (Even). This triplet does not have an odd digit in the middle position. We skip this case.
• 8 9 2: Current digit is 9 (Odd). Preceded by 8 (Even). Followed by 2 (Even). Fails because both the preceding and following digits are Even.
• 9 2 8: The current digit is 2 (Even). Skip.
• 2 8 7: The current digit is 8 (Even). Skip.
• 8 7 1: Current digit is 7 (Odd). Preceded by 8 (Even). Followed by 1 (Odd). Fails because the preceding digit is Even.
• 7 1 6: Current digit is 1 (Odd). Preceded by 7 (Odd). Followed by 6 (Even). Fails because the following digit is Even.
• 1 6 8: The current digit is 6 (Even). Skip.
• 6 8 2: The current digit is 8 (Even). Skip.
• 8 2 9: The current digit is 2 (Even). Skip.
• 2 9 8: Current digit is 9 (Odd). Preceded by 2 (Even). Followed by 8 (Even). Fails because both the preceding and following digits are Even.
• 9 8 5: The current digit is 8 (Even). Skip.
• 8 5 2: Current digit is 5 (Odd). Preceded by 8 (Even). Followed by 2 (Even). Fails because both the preceding and following digits are Even.
• 5 2 8: The current digit is 2 (Even). Skip.
• 2 8 2: The current digit is 8 (Even). Skip.
• 8 2 6: The current digit is 2 (Even). Skip.
• 2 6 4: The current digit is 6 (Even). Skip.
• 6 4 4: The current digit is 4 (Even). Skip.
• 4 4 7: The current digit is 4 (Even). Skip.
• 4 7 8: Current digit is 7 (Odd). Preceded by 4 (Even). Followed by 8 (Even). Fails because both the preceding and following digits are Even.

Note: The last digit '8' cannot be evaluated as a 'current digit' followed by another digit, as there is no digit following it.

Final Conclusion

After carefully examining the entire number series according to the specified conditions, we found no instances where an odd digit is immediately preceded by an odd digit AND immediately followed by an odd digit.

Therefore, the count of such odd digits is zero.