Putting Dots In Triangle

8-queen problem was first posed by Gauss, it’s one of typical example of backtrack algorithm. Then, this question expands for n-queen problem. In1969, Hoffman, Loessl and Moore give the arrange method of n X n chessboard problem. My research is the most number can be put in the board when the board derivative for triangle board.Research question of this paper is:give a right-angled triangle of squares in a grid whose horizontal and vertical are n squares long, let N(n) denote the maximum number of dots that can be placed into the cells of the triangle such that each row, each column, each diagonal parallel to the long side of the triangle, and each diagonal parallel to the short side of the triangle contains at most one dot. It has been proved that: when n=3t+1, N(3t+1)=2t+1(t=0,2(mod3));2t≤N(3t+1)≤2t+1(t≡1(mod3), and gives some examples of computer search. When n=3t, N(3t)=2t (t≡0,2(mod3));2t-1≤N(3t)≤2t (t≡1(mod3), and gives some examples of computer search. When n=3t+2, N(3t+2)=2t+1.There are four chapters in the thesis:In the first chapter, we will introduce the background of n queen question, the definition and known results.In the second chapter, give the proof of the layout of Brook with inequality method in the triangle board.In the third chapter, give the research of the layout of Queen in the triangle board, which is divided into n=3t,3t+1,3t+2three cases to discuss. For n=3t(t≡1(mod3)) or3t+1(t≡1(mod3)) which is not given the result has also a range and small examples of computer search.In the fourth chapter, we will present the conclusions.