3101: N皇后

3101: N皇后

Time Limit: 10 Sec  Memory Limit: 128 MBSec  Special Judge
Submit: 88  Solved: 41
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Description

n*n的棋盘,在上面摆下n个皇后，使其两两间不能相互攻击…

Input

一个数n

 

Output

第i行表示在第i行第几列放置皇后

 

Sample Input

4

Sample Output

2
4
1
3

HINT

100%的数据3<n<1000000。输出任意一种合法解即可

Source

题解：一道神(dou)奇(bi)的题目，传说中貌似有种O(N)构造N皇后解的方法，具体为啥貌似也查不到，求神犇给出证明orzorzorz（引自N皇后的构造解法）

一、当n mod 6 != 2 或 n mod 6 != 3时，有一个解为：

2,4,6,8,...,n,1,3,5,7,...,n-1       (n为偶数)

2,4,6,8,...,n-1,1,3,5,7,...,n       (n为奇数)

(上面序列第i个数为ai，表示在第i行ai列放一个皇后；... 省略的序列中，相邻两数以2递增。下同)

二、当n mod 6 == 2 或 n mod 6 == 3时，

(当n为偶数,k=n/2；当n为奇数,k=(n-1)/2)

k,k+2,k+4,...,n,2,4,...,k-2,k+3,k+5,...,n-1,1,3,5,...,k+1         (k为偶数,n为偶数)

k,k+2,k+4,...,n-1,2,4,...,k-2,k+3,k+5,...,n-2,1,3,5,...,k+1,n       (k为偶数,n为奇数)

k,k+2,k+4,...,n-1,1,3,5,...,k-2,k+3,...,n,2,4,...,k+1               (k为奇数,n为偶数)

k,k+2,k+4,...,n-2,1,3,5,...,k-2,k+3,...,n-1,2,4,...,k+1,n           (k为奇数,n为奇数)

然后就是码代码了= = 

 

/**************************************************************
    Problem: 3101
    User: HansBug
    Language: Pascal
    Result: Accepted
    Time:1832 ms
    Memory:224 kb
****************************************************************/
 
var
   i,j,k,l,m,n:longint;
begin
     readln(n);
     case n mod 6 of
          2,3:begin
                   k:=n div 2;
                   case (k mod 2)+(n mod 2)*2 of
                        0:begin
                               for i:=0 to (n-k) div 2 do writeln(k+i*2);
                               for i:=0 to (k-4) div 2 do writeln(2+i*2);
                               for i:=0 to (n-k-4) div 2 do writeln(k+3+i*2);
                               for i:=0 to k div 2 do writeln(1+2*i);
                        end;
                        2:begin
                               for i:=0 to (n-k-1) div 2 do writeln(k+i*2);
                               for i:=0 to (k-4) div 2 do writeln(2+i*2);
                               for i:=0 to (n-k-5) div 2 do writeln(k+3+i*2);
                               for i:=0 to k div 2 do writeln(1+2*i);
                               writeln(n);
                        end;
                        1:begin
                               for i:=0 to (n-k-1) div 2 do writeln(k+i*2);
                               for i:=0 to (k-3) div 2 do writeln(1+i*2);
                               for i:=0 to (n-k-3) div 2 do writeln(k+3+i*2);
                               for i:=0 to (k-1) div 2 do writeln(2+2*i);
                        end;
                        3:begin
                               for i:=0 to (n-k-2) div 2 do writeln(k+i*2);
                               for i:=0 to (k-3) div 2 do writeln(1+i*2);
                               for i:=0 to (n-k-4) div 2 do writeln(k+3+i*2);
                               for i:=0 to (k-1) div 2 do writeln(2+2*i);
                               writeln(n);
                        end;
                   end;
          end;
          else begin
               if odd(n) then
                  begin
                       for i:=1 to (n-1) div 2 do writeln(i*2);
                       for i:=1 to (n+1) div 2 do writeln(i*2-1);
                  end
               else
                   begin
                        for i:=1 to n div 2 do writeln(i*2);
                        for i:=1 to n div 2 do writeln(i*2-1);
                   end;
          end;
     end;
     readln;
end.