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Math Magic Archive

Problem of the Month (March 2015)

Consider putting the letters A and B on a square grid so that each A is adjacent to exactly m B's, and each B is adjacent to exactly n A's. ("Adjacent" means horizontally, vertically, or diagonally adjacent.) What are the smallest arrangements that work, for various values of m and n? ("Smallest" means smallest number of letters, or in the smallest bounding box.) We can extend this problem to A's touching m B's, B's touching n C's, and C's touching p A's, or to even more letters.

We can also use other definitions of adjacent. Instead of king moves, we could use moves of another piece with 8 symmetric moves: a knight, congo elephant (which moves 1 or 2 squares horizontally or vertically), phoenix (which moves 1 square horizontally or vertically or 2 squares diagonally), frog (which moves 3 squares horizontally or vertically or 1 square diagonally), or zebra (which moves 2 squares horizontally or vertically, and 3 squares in a perpendicular direction).


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