Let's analyze the positions of the pigeons to determine the total number in the row.

Pigeon P1 Position Analysis

P1 is the 9^th pigeon from the left end. This means there are 8 pigeons to the left of P1.

• Pigeons to the left of P1 = $9 - 1 = 8$

Pigeon P2 Position Analysis

P2 is the 12^th pigeon from the right end. This means there are 11 pigeons to the right of P2.

• Pigeons to the right of P2 = $12 - 1 = 11$

Pigeons Between P1 and P2

The question states there are 3 pigeons positioned between P1 and P2.

Calculating Possible Total Pigeons

There are two possible arrangements for P1 and P2:

Scenario 1: P1 is to the left of P2

The arrangement looks like this:

[Pigeons Left of P1] P1 [Pigeons Between P1 & P2] P2 [Pigeons Right of P2]

Total Pigeons = (Pigeons left of P1) + P1 + (Pigeons between P1 & P2) + P2 + (Pigeons right of P2)
Total Pigeons = $8 + 1 + 3 + 1 + 11 = 24$

Let's check if this is consistent:

• If Total = 24, P1 is 9^th from left (Correct).
• P2's position from the left end = (Pigeons left of P1) + P1 + (Pigeons between P1 & P2) + P2 = $8 + 1 + 3 + 1 = 13^{th}$
• P2's position from the right end = Total - (Position from left) + 1 = $24 - 13 + 1 = 12^{th}$. This matches the given information.

So, 24 is a possible total.

Scenario 2: P2 is to the left of P1

The arrangement looks like this:

[Pigeons Left of P2] P2 [Pigeons Between P2 & P1] P1 [Pigeons Right of P1]

We know 8 pigeons are to the left of P1. This section must include P2 and the 3 pigeons between P2 and P1.

Pigeons left of P1 = (Pigeons left of P2) + P2 + (Pigeons between P2 & P1)
$8 = (\text{Pigeons left of P2}) + 1 + 3$
Pigeons left of P2 = $8 - 4 = 4$

Now, we calculate the total number of pigeons using P2's position from the right end:

Total Pigeons = (Pigeons left of P2) + P2 + (Pigeons to the right of P2)
Total Pigeons = $4 + 1 + 11 = 16$

Let's check if this is consistent:

• If Total = 16, P2 is 12^th from the right (Correct).
• P2's position from the left end = Total - (Position from right) + 1 = $16 - 12 + 1 = 5^{th}$.
• P1's position from the left end is given as 9^th.
• Number of pigeons between P1 (9^th left) and P2 (5^th left) = $9 - 5 - 1 = 3$. This matches the given information.

So, 16 is also a possible total.

Analysis of P1, P3, and P4 Condition

The condition states that P1 and P3 are equidistant to P4. This means P4 is located exactly in the middle of P1 and P3, mathematically:

This implies that P4 is the midpoint. For P4 to be a specific pigeon (an integer position), the sum of the positions of P1 and P3 must be an even number:

This requires $\text{position}(P1)$ and $\text{position}(P3)$ to have the same parity (both must be even or both must be odd).

We know $\text{position}(P1)$ from the left is 9, which is an odd number. Therefore, $\text{position}(P3)$ must also be an odd number for P4 to be equidistant.

However, this condition does not provide P3's specific position or relate it to P1 or P2 in a way that helps eliminate either the N=16 or N=24 possibility. We don't know where P3 is, only that its position must be odd.

Conclusion on Total Pigeons

Since we found two different possible values for the total number of pigeons (16 and 24) based on the positions of P1 and P2 and the pigeons between them, and the information about P3 and P4 does not resolve this ambiguity, the total number of pigeons cannot be determined from the information given.

Final Answer

The total number of pigeons in the row cannot be determined.