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 Boosted board (Posted on 2012-02-07)
Consider an infinite chessboard. Each square contains either a 1 or an X in some pattern. (X can be any real number but for a given board, all the X's are the same.)

Each square with an X on it has weight equal to zero.
Each square with a 1 on it has a weight of 1 + N*X where N is the total number of X's on the 8 surrounding squares.

For a given value of X, find a way of tiling the board with the highest average weight per square.

Inspired by various Tower Defense games.

 Submitted by Jer Rating: 4.0000 (1 votes) Solution: (Hide) For X ≤ 1/8 all 1's is best. It always has an average weight of 1. ...111... ...111... ...111... The following pattern has an average weight of (8X+8)/9 which is tied for the best only when X=1/8 weight of 1 (it is never the sole maximum) ...111111... ...1X11X1... ...111111... ...111111... ...1X11X1... ...111111... From 1/8 to 1/4 the following pattern is the highest with a weight of (3+8X)/4 ...111111... ...1X1X1X... ...111111... ...1X1X1X... Above 1/4 this pattern of alternating strips of 1's and X's is the best with an average weight of (1+6X)/2: ...111111... ...XXXXXX... ...111111... ...XXXXXX... At X=1/8 precisely the first 3 are tied, at X=1/4 precisely the last two are tied.

 Subject Author Date re(3): Final Answer ?? (spoiler) broll 2012-02-10 09:16:59 re(2): Final Answer ?? (spoiler) Steve Herman 2012-02-09 22:31:31 re: Final Answer ?? (spoiler) Jer 2012-02-09 21:27:41 Final Answer ?? (spoiler) Steve Herman 2012-02-09 16:50:03 re: Final Answer? (spoiler) Jer 2012-02-09 10:47:34 re(2): Shaky ground? You misunderstand. brianjn 2012-02-08 21:01:28 re: Shaky ground? Steve Herman 2012-02-08 20:25:54 Final Answer? (spoiler) Steve Herman 2012-02-08 19:36:15 X = 1 (spoiler) Steve Herman 2012-02-08 09:23:44 re: Shaky ground? You misunderstand. Jer 2012-02-08 09:02:35 Shaky ground? brianjn 2012-02-08 05:51:45

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