臺灣國際科展

跛腳皇后

科展類別
臺灣國際科展
屆次
2008年
科別
數學科
學校名稱
國立臺中第一高級中學
指導老師
柯建彰
作者
王建詒、呂季桓
關鍵字
跛腳 皇后

摘要或動機

高斯曾經提出八皇后問題:八個皇后在8 × 8 的棋盤上有幾種放法可以使任意兩皇后不會互相攻擊?我們在原來的問題上加上一些條件,改變皇后攻擊規則,使得皇后失去一條對角線的攻擊方向,稱之為「跛腳皇后」。我們稱一個在棋盤上放置最多跛腳皇后使其不互相攻擊的放法為好放法;研究跛腳皇后放置在各類棋盤上其好放法的個數和性質。我們分別在六種棋盤上做討論:(1) 在平面n x n 棋盤上,我們證明了其好放法與完美極致史考倫型數列之間的對應關係,並歸納出相關的性質和定理。(2) 在平面m x n 棋盤上,我們固定一邊長度n,做出n 較小時好放法數的通式;我們也將其好放法對應至廣義史考倫。(3) 在環面n x n 棋盤上,我們說明了其好放法與完全剩餘系排列之間的對應關係,並歸納出相關的性質和定理。(4) 在環面m x n 棋盤上,我們固定gcd(m,n),做出gcd(m,n)較小時好放法數的通式。(5) 在柱面n x n 棋盤上,我們證明其與環面n x n 棋盤等價,說明其好放法具有和環面n x n 棋盤好放法相同的性質和定理。(6) 在柱面m x n 棋盤上,分成左右柱面以及上下柱面來做討論。我們歸納出相關性質和定理;並固定一邊長度n,做出n 較小時好放法數的通式。Gauss had researched about putting eight queens on the chessboard on the way that doesn’t make any queen attacks another one. Thus, we added some rules on the question: the queen loses one diagonal attacking-way and become the “lame queen”. We call a way that doesn’t make any lame queen attacks another one “a good way”. We have been investigating the amount and properties of good ways based on six kinds of chessboard: (1)We found the correspondences between the “good way” on n × n plane chessboard and the Perfect extremal Skolem-type sequence, and concluded some associated properties and theorems. (2) On m× n plane chessboard, we fixed the length n of one side of the chessboard, and accomplished the amount of good ways when n is small. We also correspond the good way to the Generalized Skolem.(3)We found the correspondences between the “good way” on n × n torus chessboard and the arrayal of complete residue system, and concluded some associated properties and theorems.(4)On m× n torus chessboard, we fixed the gcd(m,n) (greatest common divisor of m and n), and accomplished the amount of good ways when gcd(m,n) is small.(5)On n × n cylinder chessboard, we proved that this kind of chessboard is equal to torus chessboard. So the good ways, characters, and theorems on cylinder chessboard are the same as on the torus one.(6)On m× n cylinder chessboard, we separate it into two cases: left-right cylinder chessboard and up-down cylinder chessboard. We concluded some associated properties and theorems, and we also fixed the length n of one side of the chessboard and accomplished the amount of\r
the good ways when n is small.


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