2022年幻方解法整理归纳 .pdf

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1、学习资料收集于网络，仅供参考学习资料在一个由若干个排列整齐的数组成的正方形中，图中任意一横行、一纵行及对角线的几个数之和都相等，具有这种性质的图表，称为“幻方”。我国古代称为“河图”、“洛书”，又叫“纵横图”。1、奇数阶幻方罗伯特法(也有人称之为楼梯法)（如 图一：以五阶幻方为例）奇数阶幻方n 为奇数(n=3，5，7，9，11)(n=2 k+1，k=1，2，3，4，5)奇数阶幻方最经典的填法是罗伯特法(也有人称之为楼梯法)。填写方法是这样：把 1(或最小的数)放在第一行正中；按以下规律排列剩下的n n-1 个数：(1)每一个数放在前一个数的右上一格；(2)如果这个数所要放的格已经超出了顶行那么

2、就把它放在底行，仍然要放在右一列；(3)如果这个数所要放的格已经超出了最右列那么就把它放在最左列，仍然要放在上一行；(4)如果这个数所要放的格已经超出了顶行且超出了最右列，那么就把它放在前一个数的下一行同一列的格内；(5)如果这个数所要放的格已经有数填入，处理方法同(4)。这种写法总是先向“右上”的方向，象是在爬楼梯。口诀：1 居首行正中央,依次右上莫相忘上出格时往下放,右出格时往左放.排重便往自下放,右上出格一个样图一2、单偶数阶幻方122 mn分区调换法（如图二：以六阶幻方为例）把122mn阶的幻方均分成4 个同样的小幻方A、B、C、D(如图二)学习资料收集于网络，仅供参考学习资料图二（注

3、意A、B、C、D 的相对位置不能改变，因为12 m为奇数，所以A、B、C、D 均为奇数阶幻方）用连续摆数法在A 中填入21a构成幻方，同理，在B 中填入2221aa、在C中填入22312aa、在 D 中填入22413aa均构成幻方（2na）(如图三)图三（因为12 m为奇数，所以A、B、C、D 均为奇数阶幻方，必然可以用连续摆数法构造幻方）在 A 的中间一行上从左侧的第二列起取m个方格，在其它行上则从左侧第一列起取m个方格，把这些方格中的数与 D 中相应方格中的数字对调(如图四)：图四不管是几阶幻方，在A 中取数时都要从中间一行的左侧第二列开始；因为当6n时，1m，所以本例中只取了一个数）在

4、A 中从最右一列起在各行中取1m个方格，把这些方格中的数与D 中相应方格中的数字对调。（如图五）图五文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10

5、V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码

6、:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2

7、X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2

8、 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5

9、C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9

10、 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6学习资料收集于网络，仅供参考学习资料3、双偶数阶幻方mn4轴对称法（

11、如图三：以八阶幻方为例）把mn4阶的幻方均分成4 个同样的小幻方（如图六）图六 在左上角的小幻方每行每列中任取一半的方格加上底色（以便于区分），然后以轴对称的形式在其它三个小幻方中标出方格（如图七）图七（正确理解“每行每列中任取一半的方格”。本例中因为4m，所以在每个小幻方的每行每列上均取2个方格）从左上角的方格开始，按从左到右、从上到下的次序将1 64 从小到大依次填入n阶幻方，遇到有底色的方格跳过，计数，这样填满了没有底色的方格（如图八）图八（从左上角开始按从左到右、从上到下的次序将1 64 从小到大依次填入n阶幻方，当遇到有底色的方文档编码:CM8N2X2D2J2 HS5Z5C1P4X9

12、 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F1

13、0R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10

14、V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码

15、:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2

16、X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2

17、 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5

18、C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6学习资料收集于网络，仅供参考学习资料格时空出不填即可）从右下角的方格开始，按从右到左、从下到上的次序将剩下的数从小到大依次填入n阶幻方，这样填满了有底色的方格(如 图九)图九即为所求幻方。图九或者对于 n=4k 阶幻方，我们

19、先把数字按顺序填写。写好后，按4*4 把它划分成k*k 个方阵。因为n 是 4 的倍数，一定能用 4*4 的小方阵分割。然后把每个小方阵的对角线，象制作4 阶幻方的方法一样，对角线上的数字换成互补的数字，就构成幻方。（图中红色数字可用中心对称得到）文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8

20、N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2

21、J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5

22、Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4

23、X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1

24、F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L

25、10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档

26、编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6学习资料收集于网络，仅供参考学习资料文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1

27、F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L

28、10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档

29、编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8

30、N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2

31、J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6文档编码:CM8N2X2D2J2 HS5Z5C1P4X9 ZT1F10R1L10V6