幻图(一)

齐  放

问题1:请在如下括号中填入1-7,使得每个角上的数与它相邻的三个数之和等于一个定值.

这是一种环值幻图,我未在别的资料上见过,由于不好画,所以用如下表达式来代表某个幻图:

例如: H(3,1)={S=13|7,3,5,4,6,2} 代表第一图,H(3,1)={S=14|7,4,3,6,5,2}代表第二图,
H(3,1)={S=15|2,7,4,3,6,5}代表第三图.
又如: H(4,2)={S=16|9,1,8,5,6,3,7,4} 代表

其他的还有:

H(3,1)={S=16|4,6,2,7,3,5}

H(3,2)={S=14|7,4,3,5,6,1}

H(3,2)={S=16|4,3,5,6,1,7}

H(3,3)={S=15|7,4,2,6,5,1}

H(3,3)={S=16|6,5,1,7,4,2}

H(3,4)={S=14|7,1,6,3,5,2}

H(3,4)={S=18|3,5,2,7,1,6}

H(3,5)={S=16|7,1,4,6,2,3}

H(3,5)={S=17|6,4,1,7,3,2}

H(3,6)={S=16|4,5,3,2,7,1}

H(3,6)={S=18|1,4,5,3,2,7}

H(3,7)={S=16|6,1,5,3,4,2}

H(3,7)={S=17|6,1,4,5,2,3}

H(3,7)={S=18|1,4,5,2,3,6}

H(3,7)={S=19|1,5,3,4,2,6}

H(4,2)={S=16|9,1,8,5,6,3,7,4}

H(4,3)={S=21|1,9,2,7,5,6,4,8}

H(4,4)={S=17|7,1,9,3,8,2,6,5}

H(4,6)={S=23|3,9,1,7,2,8,4,5}

H(4,7)={S=19|9,1,8,3,5,4,6,2}

H(4,8)={S=24|1,9,2,5,4,7,3,6}

H(5,1)={S=18|11,2,10,5,9,3,8,6,7,4}

H(5,2)={S=22|3,8,7,5,11,4,6,10,1,9}

H(5,6)={S=21|11,1,10,4,9,2,8,5,7,3}

H(5,6)={S=27|1,11,2,8,3,10,4,7,5,9}

H(5,6)={S=23|11,2,8,7,1,9,5,3,10,4}

H(5,6)={S=25|1,10,4,5,11,3,7,9,2,8}

H(5,9)={S=27|4,8,7,3,10,5,2,11,1,6}

H(6,3)={S=22|13,1,12,6,11,2,9,8,7,4,10,5}

H(6,5)={S=28|1,12,2,9,3,11,8,4,13,6,7,10}

H(6,9)={S=28|13,2,12,5,11,3,6,10,1,8,7,4}

H(7,8)={S=36|1,15,2,11,3,14,4,10,5,13,6,9,7,12}

H(7,8)={S=28|9,7,10,3,11,6,12,2,13,5,14,1,15,4}

H(8,7)={S=30|10,4,11,8,12,3,14,6,15,2,16,5,17,1,13,9}

H(8,11)={S=42|1,13,2,16,3,12,4,15,6,10,7,14,8,9,5,17}

H(9,10)={S=45|1,19,2,14,3,18,4,13,5,17,6,12,7,16,8,11,9,15}

...

关于幻和定值,这里只能给出几种特殊情况下的公式:

以下假设N为阶数,C为中心数.
1.N是奇数,C=(N+1),1->N不在角上.
NS=NC+2(1+2+...+C)-2C+(C+1+...+C+N)
  =(N-2)C+C(C+1)+NC+N(N+1)/2
  =7N(N+1)/2
S=7(N+1)/2
2.N是奇数,C=(N+1),1->N在角上.
NS=NC+2(N+2+...+2N+1)+(1+2+...+N)
  =NC+2(N(N+1)+N(N+1)/2)+N(N+1)/2
  =N(N+1)+3(N+1)+N(N+1)/2
  =9N(N+1)/2
S=9(N+1)/2

注意到两种和数之和等于8(N+1),这是一种共轭现象,即若H(N,C)={S=S1|A(1),A(2),...A(2N)}为一个幻图,
则H(N,2N+2-C)={S=8(N+1)-S1|2N+2-A(1),2N+2-A(2),...2N+2-A(2N)}也是一个幻图.

以下给出一种奇数阶幻图的排列法:

比如要排列11阶幻图,中心数是12,先写出
H(11,12)={S=9(11+1)/2|1,( ),2,( ),3,( ),4,( ),5,( ),6,( ),7,( ),8,( ),9,( ),10,( ),11,( )}
再照如下进行即可:
H(11,12)={S=54|1,( ),2,( ),3,( ),4,( ),5,( ),6,( ),7,( ),8,( ),9,( ),10,(13),11,( )}
H(11,12)={S=54|1,( ),2,( ),3,( ),4,( ),5,( ),6,( ),7,( ),8,(14),9,( ),10,(13),11,( )}
H(11,12)={S=54|1,( ),2,( ),3,( ),4,( ),5,( ),6,(15),7,( ),8,(14),9,( ),10,(13),11,( )}
H(11,12)={S=54|1,( ),2,( ),3,( ),4,(16),5,( ),6,(15),7,( ),8,(14),9,( ),10,(13),11,( )}
H(11,12)={S=54|1,( ),2,(17),3,( ),4,(16),5,( ),6,(15),7,( ),8,(14),9,( ),10,(13),11,( )}
H(11,12)={S=54|1,( ),2,(17),3,( ),4,(16),5,( ),6,(15),7,( ),8,(14),9,( ),10,(13),11,(18)}
H(11,12)={S=54|1,( ),2,(17),3,( ),4,(16),5,( ),6,(15),7,( ),8,(14),9,(19),10,(13),11,(18)}
H(11,12)={S=54|1,( ),2,(17),3,( ),4,(16),5,( ),6,(15),7,(20),8,(14),9,(19),10,(13),11,(18)}
H(11,12)={S=54|1,( ),2,(17),3,( ),4,(16),5,(21),6,(15),7,(20),8,(14),9,(19),10,(13),11,(18)}
H(11,12)={S=54|1,( ),2,(17),3,(22),4,(16),5,(21),6,(15),7,(20),8,(14),9,(19),10,(13),11,(18)}
H(11,12)={S=54|1,(23),2,(17),3,(22),4,(16),5,(21),6,(15),7,(20),8,(14),9,(19),10,(13),11,(18)}
最后H(11,12)={S=54|1,23,2,17,3,22,4,16,5,21,6,15,7,20,8,14,9,19,10,13,11,18}就是一个幻图.
用24减去各项元素,可得到其共轭幻图:
H(11,12)={S=54|23,1,22,7,21,2,20,8,19,3,18,9,17,4,16,10,15,5,14,11,13,6}.

2001.3.20