Let the variable k range over the positive integers 1, 2, 3, ... Prove that every positive integer is in exactly one of the sets: {Floor(k * Sqrt(2)), Floor(k * (2 + Sqrt(2))}.

Draw a sprouts position whose string representation is: "1ab.c.#ba.#de1e.#" Note that "." is the end of boundary mark and "#" is the end of region mark. Try to make your position be one that could actually occur in a game of sprouts starting from a set of separate spots.

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Danny Sleator Last modified: Wed Feb 23 18:09:09 2005