Refer to the following number series and answer the question that follows. All numbers are single-digit numbers only. Counting to be done from left to right only.
(Left) 4 6 5 8 2 2 1 8 2 9 8 4 7 4 8 2 4 3 1 4 1 2 7 2 6 (Right)
How many such numbers are there, each of which is immediately preceded by a perfect square and also immediately followed by a perfect square?
(Note: 1 is also a perfect square.)

The correct option is Four.

Step 1: Understand the given criteria

We are asked to find the total count of numbers in the given series that satisfy the following condition:

• The number is immediately preceded by a perfect square (number to its left is a perfect square).

• The number is immediately followed by a perfect square (number to its right is a perfect square).

Step 2: Identify the perfect square single-digit numbers

The single-digit perfect squares among the digits in the series are:

• $1^2=1$

• $2^2=4$

• $3^2=9$

Thus, the perfect squares are 1, 4, and 9.

Step 3: Analyze the given number series

The given number sequence from left to right is:

4 6 5 8 2 2 1 8 2 9 8 4 7 4 8 2 4 3 1 4 1 2 7 2 6

Let's examine each candidate number (excluding the first and last digits, as they don't have both preceding and following elements):

1. 6 (in 4 6 5): Preceded by 4 (Square), followed by 5 (Not Square) → No

2. 8 (in 9 8 4): Preceded by 9 (Square), followed by 4 (Square) → Match 1 (Number: 8)

3. 7 (in 4 7 4): Preceded by 4 (Square), followed by 4 (Square) → Match 2 (Number: 7)

4. 3 (in 4 3 1): Preceded by 4 (Square), followed by 1 (Square) → Match 3 (Number: 3)

5. 1 (in 1 4 1): Preceded by 4 (Square), followed by 1 (Square) → Match 4 (Number: 4 in sequence 1 4 1)

Let's double-check all occurrences of patterns [Square - Number - Square]:

• Pattern 1: 9 8 4 (preceded by 9, followed by 4)

• Pattern 2: 4 7 4 (preceded by 4, followed by 4)

• Pattern 3: 4 3 1 (preceded by 4, followed by 1)

• Pattern 4: 4 1 4 (preceded by 4, followed by 4, where number is 1 in sequence ... 4 1 4 ...)

Conclusion:

There are exactly 4 such numbers that satisfy the given conditions.