Let $ m\geq n\geq 4$ be two integers. We call a $ m\times n$ board filled with 0's or 1's [i]good[/i] if

1) not all the numbers on the board are 0 or 1;

2) the sum of all the numbers in $ 3\times 3$ sub-boards is the same;

3) the sum of all the numbers in $ 4\times 4$ sub-boards is the same.

Find all $ m,n$ such that there exists a good $ m\times n$ board.
1. **Step 1: Prove that a \(6 \times 6\) board does not fit the condition.**

   - Consider a \(6 \times 6\) board. We need to calculate the sums of all \(3 \times 3\) and \(4 \times 4\) sub-boards.
   - There are \(9\) possible \(4 \times 4\) sub-boards and \(16\) possible \(3 \times 3\) sub-boards in a \(6 \times 6\) board.
   - Each number in the board is counted the same number of times in the sums of the \(3 \times 3\) and \(4 \times 4\) sub-boards.
   - Since the sum of all \(3 \times 3\) sub-boards is the same and the sum of all \(4 \times 4\) sub-boards is the same, the total sum of the \(3 \times 3\) sub-boards must be divisible by \(9\) and the total sum of the \(4 \times 4\) sub-boards must be divisible by \(16\).
   - However, it is impossible for the sum of the numbers on the board to be divisible by both \(9\) and \(16\) simultaneously, as these numbers are not coprime. Therefore, a \(6 \times 6\) board cannot be good.

2. **Step 2: Generalize to \(n \times 6\) boards for \(n \geq 6\).**

   - By the same reasoning as in Step 1, any \(n \times 6\) board for \(n \geq 6\) cannot be good because the sums of the \(3 \times 3\) and \(4 \times 4\) sub-boards would still need to be divisible by \(9\) and \(16\) respectively, which is impossible.

3. **Step 3: Provide an example of a \(n \times 5\) board for \(n \geq 5\).**

   - Consider the following \(n \times 5\) board for \(n \geq 5\):
     \[
     \begin{array}{ccccc}
     1 & 1 & 1 & 1 & 1 & \cdots \\
     1 & 1 & 1 & 1 & 1 & \cdots \\
     0 & 0 & 0 & 0 & 0 & \cdots \\
     1 & 1 & 1 & 1 & 1 & \cdots \\
     1 & 1 & 1 & 1 & 1 & \cdots \\
     \end{array}
     \]
   - This board satisfies the conditions:
     1. Not all numbers are \(0\) or \(1\).
     2. The sum of all \(3 \times 3\) sub-boards is the same.
     3. The sum of all \(4 \times 4\) sub-boards is the same.

4. **Step 4: Provide an example of a \(n \times 4\) board for \(n \geq 4\).**

   - Consider the following \(n \times 4\) board for \(n \geq 4\):
     \[
     \begin{array}{cccc}
     1 & 1 & 1 & 1 & \cdots \\
     0 & 0 & 0 & 0 & \cdots \\
     1 & 1 & 1 & 1 & \cdots \\
     1 & 1 & 1 & 1 & \cdots \\
     \end{array}
     \]
   - This board also satisfies the conditions:
     1. Not all numbers are \(0\) or \(1\).
     2. The sum of all \(3 \times 3\) sub-boards is the same.
     3. The sum of all \(4 \times 4\) sub-boards is the same.

Conclusion:
- The possible values of \((m, n)\) such that there exists a good \(m \times n\) board are \((4 + m, 4)\) and \((5 + m, 5)\) where \(m\) is a natural number.

The final answer is \( \boxed{ (4 + m, 4) } \) and \((5 + m, 5)\) where \(m\) is a natural number.