How can you put eight queens on a chess board without causing them to threaten each other? This well-known problem can be conveniently solved by a tree search method.
First, we need to write a function to determine whether two queens on the board are threatening each other. In chess, queens can move along the board's rows, columns, and the two diagonals that pass through their positions. If we number the rows and columns, we can encode positions on the board.
Use the function threat to judge whether two positions threaten each other; the function conflict is used to judge whether the position (nm) is safe when adding a new queen to the board; the function queen lists the solutions of the eight queens problem (for 2X2 and 3X3 boards No solution).
Lisp source code
;判断两个位置(i j)和(a b)是否互相威胁
(defun threat (i j a b)
(or
(equal i a) ;同行，
(equal j b) ;同列，
(equal(- i j)(- a b)) ;西南-东北对角线.
(equal(+ i j)(+ a b)) ;西北-东南对角线
)
)
;判断向棋盘上添加一个新皇后时，位置(n m)是否是安全的
(defun conflict (n m board)
(cond
((null board) nil) ;棋盘为空时，不冲突
((or (threat n m (caar board) (cadar board))
(conflict n m (cdr board)))) ;递归调用
)
)
;列出八皇后问题的解, 棋盘大小为size X size
(defun queen(size)
(prog (n m board)
(setq n 1) ;第一行
loop_n
(setq m 1) ;第一列
loop_m
(cond ((conflict n m board) (go un_do_m))) ;检查冲突. 若冲突，则转un_do_m
(setq board (cons (list n m) board)) ;若不冲突，则在棋盘上加一个皇后
(cond((> (setq n (+ n 1)) size) ;进行下一行
(print (reverse board))) ;如n放完，打印结果
)
(go loop_n) ;去找列
un_do_n
(cond
((null board)(return 'finished)) ;所有可能情况找完
(t
(setq ;用移去最后放置的一个皇后来取消最后一个结局
m (cadar board)
n (caar board)
board (cdr board)
)
)
)
un_do_m
(cond
((> (setq m (+ m 1)) size)
(go un_do_n)) ;进行下一列
(t (go loop_m))
)
)
)
program running result
The command to run on a chessboard of size 4*4 is:
(queen 4)The result of the operation is:
The command to run on a chessboard of size 8*8 is:
(queen 8)There are a total of 92 solutions, listed as follows:
((1 1) (2 5) (3 8) (4 6) (5 3) (6 7) (7 2) (8 4))
((1 1) (2 6) (3 8) (4 3) (5 7) (6 4) (7 2) (8 5))
((1 1) (2 7) (3 4) (4 6) (5 8) (6 2) (7 5) (8 3))
((1 1) (2 7) (3 5) (4 8) (5 2) (6 4) (7 6) (8 3))
((1 2) (2 4) (3 6) (4 8) (5 3) (6 1) (7 7) (8 5))
((1 2) (2 5) (3 7) (4 1) (5 3) (6 8) (7 6) (8 4))
((1 2) (2 5) (3 7) (4 4) (5 1) (6 8) (7 6) (8 3))
((1 2) (2 6) (3 1) (4 7) (5 4) (6 8) (7 3) (8 5))
((1 2) (2 6) (3 8) (4 3) (5 1) (6 4) (7 7) (8 5))
((1 2) (2 7) (3 3) (4 6) (5 8) (6 5) (7 1) (8 4))
((1 2) (2 7) (3 5) (4 8) (5 1) (6 4) (7 6) (8 3))
((1 2) (2 8) (3 6) (4 1) (5 3) (6 5) (7 7) (8 4))
((1 3) (2 1) (3 7) (4 5) (5 8) (6 2) (7 4) (8 6))
((1 3) (2 5) (3 2) (4 8) (5 1) (6 7) (7 4) (8 6))
((1 3) (2 5) (3 2) (4 8) (5 6) (6 4) (7 7) (8 1))
((1 3) (2 5) (3 7) (4 1) (5 4) (6 2) (7 8) (8 6))
((1 3) (2 5) (3 8) (4 4) (5 1) (6 7) (7 2) (8 6))
((1 3) (2 6) (3 2) (4 5) (5 8) (6 1) (7 7) (8 4))
((1 3) (2 6) (3 2) (4 7) (5 1) (6 4) (7 8) (8 5))
((1 3) (2 6) (3 2) (4 7) (5 5) (6 1) (7 8) (8 4))
((1 3) (2 6) (3 4) (4 1) (5 8) (6 5) (7 7) (8 2))
((1 3) (2 6) (3 4) (4 2) (5 8) (6 5) (7 7) (8 1))
((1 3) (2 6) (3 8) (4 1) (5 4) (6 7) (7 5) (8 2))
((1 3) (2 6) (3 8) (4 1) (5 5) (6 7) (7 2) (8 4))
((1 3) (2 6) (3 8) (4 2) (5 4) (6 1) (7 7) (8 5))
((1 3) (2 7) (3 2) (4 8) (5 5) (6 1) (7 4) (8 6))
((1 3) (2 7) (3 2) (4 8) (5 6) (6 4) (7 1) (8 5))
((1 3) (2 8) (3 4) (4 7) (5 1) (6 6) (7 2) (8 5))
((1 4) (2 1) (3 5) (4 8) (5 2) (6 7) (7 3) (8 6))
((1 4) (2 1) (3 5) (4 8) (5 6) (6 3) (7 7) (8 2))
((1 4) (2 2) (3 5) (4 8) (5 6) (6 1) (7 3) (8 7))
((1 4) (2 2) (3 7) (4 3) (5 6) (6 8) (7 1) (8 5))
((1 4) (2 2) (3 7) (4 3) (5 6) (6 8) (7 5) (8 1))
((1 4) (2 2) (3 7) (4 5) (5 1) (6 8) (7 6) (8 3))
((1 4) (2 2) (3 8) (4 5) (5 7) (6 1) (7 3) (8 6))
((1 4) (2 2) (3 8) (4 6) (5 1) (6 3) (7 5) (8 7))
((1 4) (2 6) (3 1) (4 5) (5 2) (6 8) (7 3) (8 7))
((1 4) (2 6) (3 8) (4 2) (5 7) (6 1) (7 3) (8 5))
((1 4) (2 6) (3 8) (4 3) (5 1) (6 7) (7 5) (8 2))
((1 4) (2 7) (3 1) (4 8) (5 5) (6 2) (7 6) (8 3))
((1 4) (2 7) (3 3) (4 8) (5 2) (6 5) (7 1) (8 6))
((1 4) (2 7) (3 5) (4 2) (5 6) (6 1) (7 3) (8 8))
((1 4) (2 7) (3 5) (4 3) (5 1) (6 6) (7 8) (8 2))
((1 4) (2 8) (3 1) (4 3) (5 6) (6 2) (7 7) (8 5))
((1 4) (2 8) (3 1) (4 5) (5 7) (6 2) (7 6) (8 3))
((1 4) (2 8) (3 5) (4 3) (5 1) (6 7) (7 2) (8 6))
((1 5) (2 1) (3 4) (4 6) (5 8) (6 2) (7 7) (8 3))
((1 5) (2 1) (3 8) (4 4) (5 2) (6 7) (7 3) (8 6))
((1 5) (2 1) (3 8) (4 6) (5 3) (6 7) (7 2) (8 4))
((1 5) (2 2) (3 4) (4 6) (5 8) (6 3) (7 1) (8 7))
((1 5) (2 2) (3 4) (4 7) (5 3) (6 8) (7 6) (8 1))
((1 5) (2 2) (3 6) (4 1) (5 7) (6 4) (7 8) (8 3))
((1 5) (2 2) (3 8) (4 1) (5 4) (6 7) (7 3) (8 6))
((1 5) (2 3) (3 1) (4 6) (5 8) (6 2) (7 4) (8 7))
((1 5) (2 3) (3 1) (4 7) (5 2) (6 8) (7 6) (8 4))
((1 5) (2 3) (3 8) (4 4) (5 7) (6 1) (7 6) (8 2))
((1 5) (2 7) (3 1) (4 3) (5 8) (6 6) (7 4) (8 2))
((1 5) (2 7) (3 1) (4 4) (5 2) (6 8) (7 6) (8 3))
((1 5) (2 7) (3 2) (4 4) (5 8) (6 1) (7 3) (8 6))
((1 5) (2 7) (3 2) (4 6) (5 3) (6 1) (7 4) (8 8))
((1 5) (2 7) (3 2) (4 6) (5 3) (6 1) (7 8) (8 4))
((1 5) (2 7) (3 4) (4 1) (5 3) (6 8) (7 6) (8 2))
((1 5) (2 8) (3 4) (4 1) (5 3) (6 6) (7 2) (8 7))
((1 5) (2 8) (3 4) (4 1) (5 7) (6 2) (7 6) (8 3))
((1 6) (2 1) (3 5) (4 2) (5 8) (6 3) (7 7) (8 4))
((1 6) (2 2) (3 7) (4 1) (5 3) (6 5) (7 8) (8 4))
((1 6) (2 2) (3 7) (4 1) (5 4) (6 8) (7 5) (8 3))
((1 6) (2 3) (3 1) (4 7) (5 5) (6 8) (7 2) (8 4))
((1 6) (2 3) (3 1) (4 8) (5 4) (6 2) (7 7) (8 5))
((1 6) (2 3) (3 1) (4 8) (5 5) (6 2) (7 4) (8 7))
((1 6) (2 3) (3 5) (4 7) (5 1) (6 4) (7 2) (8 8))
((1 6) (2 3) (3 5) (4 8) (5 1) (6 4) (7 2) (8 7))
((1 6) (2 3) (3 7) (4 2) (5 4) (6 8) (7 1) (8 5))
((1 6) (2 3) (3 7) (4 2) (5 8) (6 5) (7 1) (8 4))
((1 6) (2 3) (3 7) (4 4) (5 1) (6 8) (7 2) (8 5))
((1 6) (2 4) (3 1) (4 5) (5 8) (6 2) (7 7) (8 3))
((1 6) (2 4) (3 2) (4 8) (5 5) (6 7) (7 1) (8 3))
((1 6) (2 4) (3 7) (4 1) (5 3) (6 5) (7 2) (8 8))
((1 6) (2 4) (3 7) (4 1) (5 8) (6 2) (7 5) (8 3))
((1 6) (2 8) (3 2) (4 4) (5 1) (6 7) (7 5) (8 3))
((1 7) (2 1) (3 3) (4 8) (5 6) (6 4) (7 2) (8 5))
((1 7) (2 2) (3 4) (4 1) (5 8) (6 5) (7 3) (8 6))
((1 7) (2 2) (3 6) (4 3) (5 1) (6 4) (7 8) (8 5))
((1 7) (2 3) (3 1) (4 6) (5 8) (6 5) (7 2) (8 4))
((1 7) (2 3) (3 8) (4 2) (5 5) (6 1) (7 6) (8 4))
((1 7) (2 4) (3 2) (4 5) (5 8) (6 1) (7 3) (8 6))
((1 7) (2 4) (3 2) (4 8) (5 6) (6 1) (7 3) (8 5))
((1 7) (2 5) (3 3) (4 1) (5 6) (6 8) (7 2) (8 4))
((1 8) (2 2) (3 4) (4 1) (5 7) (6 5) (7 3) (8 6))
((1 8) (2 2) (3 5) (4 3) (5 1) (6 7) (7 4) (8 6))
((1 8) (2 3) (3 1) (4 6) (5 2) (6 5) (7 7) (8 4))
((1 8) (2 4) (3 1) (4 3) (5 6) (6 2) (7 7) (8 5))