Question 82
The columns and the rows of a $3n \times 3n$ square board are numbered $1, 2, \ldots, 3n$. Every square $(x, y)$ with $1 \leq x, y \leq 3n$ is colored asparagus, byzantium or citrine according as the modulo $3$ remainder of $x+y$ is $0$, $1$ or $2$ respectively. One token colored asparagus, byzantium or citrine is placed on each square, so that there are $3n^{2}$ tokens of each color.
Suppose that one can permute the tokens so that each token is moved to a distance of at most $d$ from its original position, each asparagus token replaces a byzantium token, each byzantium token replaces a citrine token, and each citrine token replaces an asparagus token. Prove that it is possible to permute the tokens so that each token is moved to a distance of at most $d+2$ from its original position, and each square contains a token with the same color as the square.
Suppose that one can permute the tokens so that each token is moved to a distance of at most $d$ from its original position, each asparagus token replaces a byzantium token, each byzantium token replaces a citrine token, and each citrine token replaces an asparagus token. Prove that it is possible to permute the tokens so that each token is moved to a distance of at most $d+2$ from its original position, and each square contains a token with the same color as the square.