MathDB

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
[*]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. \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.
[*]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?

Problem Statement