Consider a grid with $n{}$ lines and $m{}$ columns $(n,m\in\mathbb{N},m,n\ge2)$ made of $n\cdot m \; 1\times1$ squares called ${cells}$. A ${snake}$ is a sequence of cells with the following properties: the first cell is on the first line of the grid and the last cell is on the last line of the grid, starting with the second cell each has a common side with the previous cell and is not above the previous cell. Define the ${length}$ of a snake as the number of cells it's made of. Find the arithmetic mean of the lengths of all the snakes from the grid.   
To find the arithmetic mean of the lengths of all the snakes in the grid, we need to consider the expected value of the length of a snake. Let's break down the problem step by step.

1. **Understanding the Snake Movement**:
   - A snake starts at any cell in the first row and ends at any cell in the last row.
   - Each cell in the snake must share a side with the previous cell and must not be above the previous cell.

2. **Expected Length Calculation**:
   - Let's denote the length of a snake by \( L \).
   - We need to calculate the expected value \( E(L) \) of the length of a snake.

3. **Expected Number of Moves in Each Row**:
   - When a snake drops down to a new row, it can move left or right until it reaches the edge of the grid.
   - If a snake drops to the leftmost cell of a row, the expected number of cells it will cover in that row is \( \frac{1 + 2 + \ldots + m}{m} = \frac{m+1}{2} \).
   - If a snake drops to the second cell from the left, the expected number of cells it will cover in that row is \( \frac{(m-1) + (m-2) + \ldots + 1}{m-1} = \frac{m}{2} \).

4. **General Case**:
   - For a general case, if a snake drops to the \( x \)-th cell from the left, the expected number of cells it will cover in that row is:
     \[
     \frac{\frac{x(x+1)}{2} + \frac{(m-x)(m-x+1)}{2}}{m}
     \]
   - Simplifying this expression, we get:
     \[
     \frac{\frac{x(x+1) + (m-x)(m-x+1)}{2}}{m}
     \]

5. **Summing Up the Expected Values**:
   - We need to sum up the expected values for all possible starting positions in the first row and divide by the total number of starting positions \( m \).

6. **Final Calculation**:
   - The expected length of a snake is the sum of the expected values for each row, starting from the first row to the last row.
   - Since the snake must traverse \( n \) rows, the expected length \( E(L) \) is:
     \[
     E(L) = n \cdot \frac{m+1}{2}
     \]

The final answer is \( \boxed{n \cdot \frac{m+1}{2}} \).