幻方问题（Magic Squares）

news2024/9/23 5:16:44

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")