The columns and the row of a 3n×3n square board are numbered 1,2,…,3n. Every square (x,y) with 1≤x,y≤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 3n2 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. geometryrectanglegraph theorycombinatoricsIMO Shortlist