对角线模n王后覆盖问题

ON THE DIAGONAL MODULAR n-QUEENS DOMINATION PROBLEM

所谓模ｎ王后是指在ｎ×ｎ国际象棋棋盘上不仅可以横走，竖走，而且可以沿对角线（没有折断的和折断了的）方向行走的棋子。我们记可以覆盖整个ｎ×ｎ棋盘且位于主对角线上的模ｎ王后的最小个数为，且对于所有．在本文中，我们证明了：当ｎ≡２（ｍｏｄ４）时，Ｄｉａｇ（ｎ）＝ｎ／２；当ｎ≡０（ｍｏｄ４）时，Ｄｉａｇ（ｎ）＝３ｎ／４－１；当ｎ为奇数时，Ｄｉａｇ（ｎ）＝ｎ－Ｒ（ｎ）．这里，ｎ＞２。

The modular n-queens are those pieces on the n×n chessboard, which can move freely along ranks, or files, or diagonal lines(either broken, or unbroken ones). Let Diag (n) be minimal number m of modular n -queens which place on major diagonal of n×n chessboard and dominate all squares. In this paper, we prove that Diag (n) = n/2 if n≡2(mod4), Diag (n) = 3n/4 -1 if n≡0 (mod4), Diag (n) = n -R(n) if n is odd. There