Given a positive integer $N$. There are three squirrels that each have an integer. It is known that the largest integer and the least one differ by exactly $N$. Each time, the squirrel with the second largest integer looks at the squirrel with the largest integer. If the integers they have are different, then the squirrel with the second largest integer would be unhappy and attack the squirrel with the largest one, making its integer decrease by two times the difference between the two integers. If the second largest integer is the same as the least integer, only of the squirrels would attack the squirrel with the largest integer. The attack continues until the largest integer becomes the same as the second largest integer. What is the maximum total number of attacks these squirrels make?

Proposed by USJL, ST.
1. **Initial Setup**: Let the integers held by the three squirrels be \(a\), \(b\), and \(c\) such that \(a \geq b \geq c\). It is given that the difference between the largest and the smallest integer is \(N\), i.e., \(a - c = N\).

2. **Attack Mechanism**: The squirrel with the second largest integer (\(b\)) attacks the squirrel with the largest integer (\(a\)) if \(a \neq b\). The attack reduces \(a\) by \(2(a - b)\), resulting in the new integers being \(b\), \(2b - a\), and \(c\).

3. **Case Analysis**:
    - **Case 1**: If \(c \geq 2b - a\), then the new difference between the largest and smallest integers is \(b - c\), which is less than the original difference \(a - c\).
    - **Case 2**: If \(c < 2b - a\), then the new difference is \(b - (2b - a) = a - b\), which is also less than the original difference \(a - c\).

4. **Reduction of Difference**: Each attack reduces the difference between the largest and smallest integers. The process continues until the largest integer becomes equal to the second largest integer.

5. **Maximum Number of Attacks**: To find the maximum number of attacks, consider the initial setup \(a = N\), \(b = N-1\), and \(c = 0\). After each attack, the integers change as follows:
    - Initial: \(N, N-1, 0\)
    - After 1st attack: \(N-1, N-2, 0\)
    - After 2nd attack: \(N-2, N-3, 0\)
    - ...
    - After \(N\)th attack: \(0, -1, 0\)

   After \(N\) attacks, the largest integer becomes equal to the second largest integer, and the process stops.

6. **Conclusion**: The maximum total number of attacks these squirrels can make is \(N\).

The final answer is \(\boxed{N}\)