奇數魔方陣

奇數魔方陣是將 1 到奇數 n 的數字排列在 n x n 的方陣，各行、各列與各對角線的和必須相同：: 17 24 1 8 15; 23 5 7 14 16; 4 6 13 20 22; 10 12 19 21 3; 11 18 25 2 9

解法思路

奇數魔方陣最簡單，第一個數字放在第一行第一列的中央，然後向右（左）上填，如果右（左）上已有數字就向下填，如所示：

奇數魔方陣

為了計算方便，利用索引 1 到 n 的部份，在計算是向右（左）上或向下時，可以將索引值除以 n 值，如果得到餘數為 1 就向下，否則就往右（左）上，原理很簡單，看看是不是已經在同一列上繞一圈就對了。

程式實作

#include <stdio.h> 
#include <stdlib.h> 

#define N 5 

int main(void) { 
    int matrix[N][N] = {0}; 

    int i = 0; 
    int j = (N+1) / 2; 
    int key;
    for(key = 1; key <= N*N; key++) { 
        if((key % N) == 1) 
            i++; 
        else { 
            i--; 
            j++; 
        } 

        if(i == 0) 
            i = N; 
        if(j > N) 
            j = 1; 

        matrix[i - 1][j - 1] = key; 
    } 

    int m, n;
    for(m = 0; m < N; m++) { 
        for(n = 0; n < N; n++) 
            printf("%2d ", matrix[m][n]); 
    } 

    return 0; 
}

public class Matrix {
    public static int[][] magic(int n) {
        int[][] matrix = new int[n][n]; 

        for(int i = 0, j = (n + 1) / 2, key = 1; key <= n*n; key++) { 
            if((key % n) == 1) i++; 
            else { i--; j++; } 

            if(i == 0) i = n; 
            if(j > n) j = 1; 

            matrix[i - 1][j - 1] = key; 
        }
        
        return matrix;
    }
    
    public static void main(String[] args) {
        for(int[] row : Matrix.magic(5)) {
            for(int number: row) {
                System.out.printf("%2d ", number);
            }
            System.out.println();
         }
    }
}

def magic(n):
    square = []
    for i in range(n):
        square.append([0] * n)
        
    i = 0
    j = (n + 1) // 2
    for key in range(1, n ** 2 + 1):
        if key % n == 1:
            i += 1
        else:
            i -= 1
            j += 1
        if i == 0:
            i = n
        if j > n:
            j = 1
        square[i - 1][j - 1] = key

    return square

matrix = magic(5)
print(matrix)

object Matrix {
    def magic(n: Int) = {
        val matrix = new Array[Array[Int]](n, n)
        var i = 0
        var j = (n + 1) / 2
        for(key <- 1 to n * n) {
            if((key % n) == 1) i += 1
            else { i -= 1; j += 1}

            if(i == 0) i = n
            if(j > n) j = 1
            
            matrix(i - 1)(j - 1) = key
        }
        
        matrix
    }
}

Matrix.magic(5).foreach(row => {
    row.foreach(number => printf("%2d ", number))
    println()
})

def magic(n)
    matrix = Array.new(n) {
        Array.new(n, 0)
    }
    i = 0
    j = (n + 1) / 2
    1.upto(n ** 2) { |key|
        if key % n == 1
            i += 1
        else
            i -= 1
            j += 1
        end
        if i == 0
            i = n
        end
        if j > n
            j = 1
        end
        matrix[i - 1][j - 1] = key
    }

    matrix
end

matrix = magic(5)
p matrix

function magic(n) {
    let matrix = [];
    for(let i = 0; i < n; i++) {
        matrix.push([]);
    }

    let i = 0;
    let j = Math.floor((n + 1) / 2);
    for(let key = 1; key < n * n + 1; key ++) {
        if(key % n == 1) {
            i++;
        }
        else {
            i--;
            j++;
        }
        if(i == 0) {
            i = n;
        }
        if (j > n) {
            j = 1;
        }
        matrix[i - 1][j - 1] = key;
    }        
    
    return matrix
}

matrix = magic(5)
console.log(matrix)

slice :: Int -> Int -> [a] -> [a]
slice start stop xs = take (stop - start) (drop start xs)

updated i j v mx =
    let 
        row = mx !! i
        updatedRow = (slice 0 j row) ++ [v] ++ (slice (j + 1) (length row) row)
        before = slice 0 i mx
        after = slice (i + 1) (length mx) mx
    in before ++ [updatedRow] ++ after
    
magic n =
    let 
        matrix = replicate n $ replicate n 0
    in _magic 0 ((n + 1) `div` 2) 1 n matrix
    where
    _magic i j key n matrix=
        if key == n * n + 1 then matrix
        else
        let 
            (ni, nj) = nij i j key n
            nMatrix = updated (ni - 1) (nj - 1) key matrix
        in _magic ni nj (key + 1) n nMatrix
        
    nij i j key n = 
        if key `mod` n == 1 then (i + 1, j)
        else
        let 
            ni = i - 1
            nj = j + 1
        in (if ni == 0 then n else ni, if nj > n then 1 else nj)
        
main = print $ magic 5