HMMT 十一月 2016 · 冲刺赛 · 第 29 题

英文解析

29. [ 15 ] We want to design a new chess piece, the American, with the property that (i) the American cannever attack itself, and (ii) if an American A attacks another American A , then A also attacks A .
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    Let m be the number of squares that an American attacks when placed in the top left corner of an 8 by
    8 chessboard. Let n be the maximal number of Americans that can be placed on the 8 by 8 chessboardsuch that no Americans attack each other, if one American must be in the top left corner. Find thelargest possible value of mn .
    Proposed by: Kevin Yang
    Answer: 1024
    Since one of the Americans must be in the top left corner, that eliminates m squares from considerationfor placing additional Americans. So m + n is at most 64, which implies mn can be at most 1024. Toachieve 1024, we can color a chessboard the normal way, and say that an American attacks all squaresof the opposite color. Then the American in the top left corner attacks the 32 squares of the oppositecolor, and placing all Americans on the squares of the same color as the top-left corner guarantees no
    Americans attack each other.