Some frogs, some red and some others green, are going to move in an $11 \times 11$ grid, according to the following rules. If a frog is located, say, on the square marked with # in the following diagram, then
[list]
[*]If it is red, it can jump to any square marked with an x. 
[*]if it is green, it can jump to any square marked with an o.[/list]
\[\begin{tabular}{| p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | p{0.08cm} | l}
\hline
&&&&&&\\ \hline
&&x&&o&&\\ \hline
&o&&&&x&\\ \hline
&&&\small{\#}&&&\\ \hline
&x&&&&o&\\ \hline
&&o&&x&&\\ \hline
&&&&&&\\ \hline
\end{tabular}
\]
We say 2 frogs (of any color) can meet at a square if both can get to the same square in one or more jumps, not neccesarily with the same amount of jumps.
[list=a]
[*]Prove if 6 frogs are placed, then there exist at least 2 that can meet at a square.
[*]For which values of $k$ is it possible to place one green and one red frog such that they can meet at exactly $k$ squares?[/list]
### Part (a)

1. **Define the movement rules and reachable squares:**
   - A red frog can jump to any square marked with an 'x'.
   - A green frog can jump to any square marked with an 'o'.

2. **Calculate the number of reachable squares for each frog:**
   - Consider a red frog at position \((i, j)\). The squares it can jump to are:
     \[
     (i-1, j-2), (i-2, j-1), (i+1, j-2), (i+2, j-1), (i-1, j+2), (i-2, j+1), (i+1, j+2), (i+2, j+1)
     \]
   - Similarly, for a green frog at position \((i, j)\), the squares it can jump to are:
     \[
     (i-1, j-1), (i-1, j+1), (i+1, j-1), (i+1, j+1)
     \]

3. **Estimate the number of reachable squares:**
   - Each frog can reach a certain number of squares. For simplicity, assume each frog can reach at least 24 different squares (this is a rough estimate considering the grid size and possible jumps).

4. **Apply the pigeonhole principle:**
   - There are 6 frogs and each can reach at least 24 squares.
   - The total number of reachable squares is at least \(6 \times 24 = 144\).
   - Since the grid has only \(11 \times 11 = 121\) squares, by the pigeonhole principle, there must be at least one square that is reachable by at least two frogs.

Thus, there exist at least 2 frogs that can meet at a square.

### Part (b)

1. **Define the problem:**
   - We need to determine for which values of \(k\) it is possible to place one green and one red frog such that they can meet at exactly \(k\) squares.

2. **Analyze the movement patterns:**
   - A red frog can jump to 8 possible squares.
   - A green frog can jump to 4 possible squares.

3. **Determine the intersection of reachable squares:**
   - The maximum number of common squares they can meet at is the minimum of the number of squares each can reach, which is 4 (since the green frog can only reach 4 squares).

4. **Possible values of \(k\):**
   - The values of \(k\) can range from 0 to 4, depending on the placement of the frogs and the overlap of their reachable squares.

The final answer is \( \boxed{ k } \) can be any value from 0 to 4.