幻方问题（Magic Squares）
目录
基本介绍
丢勒-幻方
高阶幻方矩阵
习题
1. 幻方检测
2. durerperm
3. 颜色分配表
4. 幻方矩阵的逆矩阵
5. 幻方矩阵的秩
基本介绍
n×n幻方是含有1到n^2的整数数组，排列后是的每一行、每一列、正反两条主对角线上数字的和都是相同的。对于每个n>2都有很多不同的n×n幻方，函数magic(n)只产生一个特定的幻方。
rot90 将数组旋转 90 度
%% A Few Elementary Array Operations.
format short
A = magic(3)
sum(A)
sum(A')'
sum(diag(A))
sum(diag(flipud(A)))
sum(1:9)/3
for k = 0:3
rot90(A,k)
rot90(A',k)
end丢勒-幻方
%% Durer's Melancolia
clear all
close all
figure
load durer
whos %数组X中给出像素的值为灰度颜色分配表(olormap) map的索引值
image(X)
colormap(map)
axis image
%% Durer's Magic Square
figure
load detail
image(X)
colormap(map)
axis image
A = magic(4)
A = A(:,[1 3 2 4])
高阶幻方矩阵
幻方和
%% Magic Sum n = (3:10)'; (n.^3 + n)/2
分三种情况生成幻方矩阵：
（1）n为奇数
（2）n为单偶数（能够被2整除，但是不能被4整除）
（3）n为双偶数（能被4整除）
生成奇数和双偶数幻方矩阵
%% Odd Order 奇数阶幻方矩阵
n = 5
[I,J] = ndgrid(1:n);
A = mod(I+J+(n-3)/2,n);
B = mod(I+2*J-2,n);
M = n*A + B + 1
%% Doubly Even Order n是为双偶数
n = 4
M = reshape(1:n^2,n,n)';
[I,J] = ndgrid(1:n);
K = fix(mod(I,4)/2) == fix(mod(J,4)/2);
M(K) = n^2+1 - M(K)
%% Rank 幻方矩阵的秩
figure
for n = 3:20
r(n) = rank(magic(n));
end
bar(r)
axis([2 21 0 20])
%% Ismagical 判定是否为幻方矩阵
help ismagical
for n = 3:10
ismagical(magic(n))
end表面图
%% Surf Plots 表面图
figure
for n = 9:12
subplot(2,2,n-8)
surf(rot90(magic(n)))
axis tight off
text(0,0,20,num2str(n))
end
set(gcf,'color','white')
习题
1. 幻方检测
function result = ismagic(A)
[m, n] = size(A);
% Check if A is a square matrix
if m ~= n
error('Input matrix must be a square matrix.');
end
% Calculate the expected sum
expectedSum = sum(A(1, :));
% Check rows
rowSums = sum(A, 2);
if ~all(rowSums == expectedSum)
result = false;
return;
end
% Check columns
colSums = sum(A, 1);
if ~all(colSums == expectedSum)
result = false;
return;
end
% Check main diagonal
if sum(diag(A)) ~= expectedSum
result = false;
return;
end
% Check secondary diagonal (if matrix size is odd)
if mod(n, 2) == 1
if sum(diag(flip(A))) ~= expectedSum
result = false;
return;
end
end
% If all checks pass, the matrix is a magic square
result = true;
end
2. durerperm
换行或者换列后，元素和保持不变。
function durerperm(arg)
% DURERPERM Permute Durer's magic square.
% Click on two different rows or columns.
% Is the result still a magic square?
if nargin == 0
shg
load detail
image(X,'buttondownfcn','durerperm(''click'')')
colormap(map)
axis image off
set(gca,'userdata',[])
title('Click on two rows or columns')
elseif isequal(arg,'click')
cp = get(gca,'currentpoint');
a = 35;
b = 29;
h = 74;
w = 75;
p = [ceil((cp(1,1)-a)/h) ceil((cp(1,2)-b)/w)];
if any(p < 1) || any(p > 4), return, end
if isempty(get(gca,'userdata'))
set(gca,'userdata',p)
else
p1 = get(gca,'userdata');
p2 = p;
Xp = get(gca,'child');
X = get(Xp,'cdata');
c = h*(0:3);
d = w*(0:3);
c([p1(2) p2(2)]) = c([p2(2) p1(2)]);
d([p1(1) p2(1)]) = d([p2(1) p1(1)]);
i = 0:h-1;
j = 0:w-1;
I = a+[c(1)+i c(2)+i c(3)+i c(4)+i];
J = b+[d(1)+j d(2)+j d(3)+j d(4)+j];
K = a+(0:4*h-1);
L = b+(0:4*w-1);
X(K,L) = X(I,J);
set(Xp,'cdata',X)
set(gca,'userdata',[])
end
end
3. 颜色分配表
clear
load detail; whos
map, caption
X(101:130, 101:118)
min(min(X))
max(max(X))
image(X)
axis image
%colormap(map)
colormap("hot")
%colormap("cool")
%colormap("copper")
%colormap("pink")
4. 幻方矩阵的逆矩阵
幻方矩阵的逆矩阵可以通过inv求出
X= inv(magic(n))这个逆矩阵的元素不再保证为正数、整数，但是具有相同的行和列和。
对于偶数阶次n来说，行列数det（magic(n)）为0，矩阵为奇异矩阵，逆矩阵不存在。
5. 幻方矩阵的秩
三种不同算法产生的幻方矩阵的秩是不同的：
阶次n 秩
奇数 n
单偶数 n/2+2
双偶数 3