Question 20
Let $n$ be a positive odd integer. At each point of the $2n\times2n$ grid
$$
C=\{(i,j):i,j=0,1,\dots,2n-1\}
$$
there is a guinea pig facing one of the positive or negative $x$- or $y$-directions. Jeff wants to keep $n^2+1$ guinea pigs and remove the rest.
In every round, all remaining guinea pigs simultaneously move one unit in their fixed directions. If the intended destination $(i,j)$ is outside $C$, the animal instead moves to the unique $(p,q)\in C$ with $p\equiv i\pmod{2n}$ and $q\equiv j\pmod{2n}$. For example, a move from $(2,0)$ toward $(2,-1)$ ends at $(2,2n-1)$.
Jeff wants no two retained guinea pigs ever to have the same destination in a round, and no two ever to move to each other's starting points in a round. Prove that he can always achieve this, regardless of the initial directions.
$$
C=\{(i,j):i,j=0,1,\dots,2n-1\}
$$
there is a guinea pig facing one of the positive or negative $x$- or $y$-directions. Jeff wants to keep $n^2+1$ guinea pigs and remove the rest.
In every round, all remaining guinea pigs simultaneously move one unit in their fixed directions. If the intended destination $(i,j)$ is outside $C$, the animal instead moves to the unique $(p,q)\in C$ with $p\equiv i\pmod{2n}$ and $q\equiv j\pmod{2n}$. For example, a move from $(2,0)$ toward $(2,-1)$ ends at $(2,2n-1)$.
Jeff wants no two retained guinea pigs ever to have the same destination in a round, and no two ever to move to each other's starting points in a round. Prove that he can always achieve this, regardless of the initial directions.