Ray

마방진

10 min read

마방진은 정사각형 격자 안에 숫자를 배열하여 모든 행, 열, 대각선의 합이 동일하게 되도록 하는 숫자 배열입니다.

이러한 배열을 대충 파이썬 코드를 작성해서 찾아보면 뿐만 아니라 더 큰 수에서도 찾을 수 있으며

import numpy as np
from multiprocessing import Pool
 
# 홀수 차수의 마방진 생성
def generate_odd_magic_square(n):
    magic_square = np.zeros((n,n), dtype=int)
    num = 1
    i, j = 0, n//2
    
    while num <= n*n:
        magic_square[i, j] = num
        num += 1
        newi, newj = (i-1) % n, (j+1) % n
        if magic_square[newi, newj]:
            i += 1
        else:
            i, j = newi, newj
            
    return magic_square
 
# 4n차수의 마방진 생성
def generate_doubly_even_magic_square(n):
    arr = [[(n*y)+x+1 for x in range(n)]for y in range(n)]
    for i in range(0,n//4):
        for j in range(0,n//4):
            arr[i][j] = (n*n + 1) - arr[i][j];
    for i in range(0,n//4):
        for j in range(3 * (n//4),n):
            arr[i][j] = (n*n + 1) - arr[i][j];
    for i in range(3 * (n//4),n):
        for j in range(0,n//4):
            arr[i][j] = (n*n + 1) - arr[i][j];
    for i in range(3 * (n//4),n):
        for j in range(3 * (n//4),n):
            arr[i][j] = (n*n + 1) - arr[i][j];
    for i in range(n//4,3 * (n//4)):
        for j in range(n//4,3 * (n//4)):
            arr[i][j] = (n*n + 1) - arr[i][j];
    return np.array(arr)
 
# 4n+2차수의 마방진 생성
def generate_singly_even_magic_square(s):
    if s % 2 == 1:
        s += 1
    while s % 4 == 0:
        s += 2
    q = [[0 for j in range(s)] for i in range(s)]
    z = s // 2
    b = z * z
    c = 2 * b
    d = 3 * b
    o = generate_odd_magic_square(z)
    for j in range(0, z):
        for i in range(0, z):
            a = o[i][j]
            q[i][j] = a
            q[i + z][j + z] = a + b
            q[i + z][j] = a + c
            q[i][j + z] = a + d
    lc = z // 2
    rc = lc
    for j in range(0, z):
        for i in range(0, s):
            if i < lc or i > s - rc or (i == lc and j == lc):
                if not (i == 0 and j == lc):
                    t = q[i][j]
                    q[i][j] = q[i][j + z]
                    q[i][j + z] = t
    return np.array(q)
 
def generate_magic_square(n):
    if n % 2 == 1:
        return generate_odd_magic_square(n)
    elif n % 4 == 0:
        return generate_doubly_even_magic_square(n)
    else:
        return generate_singly_even_magic_square(n)
 
def is_valid_magic_square(magic_square):
    n = len(magic_square)
    magic_sum = np.sum(magic_square, axis=0)[0]
    return (
        all(np.sum(magic_square, axis=0) == magic_sum) and
        all(np.sum(magic_square, axis=1) == magic_sum) and
        np.sum(np.diag(magic_square)) == magic_sum and
        np.sum(np.diag(np.fliplr(magic_square))) == magic_sum
    )
 
def worker(n):
    magic_square = generate_magic_square(n)
    result_str = f'Magic Square of order {n}x{n}:\n'
    result_str += str(magic_square) + '\n'
    result_str += f'Magic sum: {np.sum(magic_square, axis=0)[0]}\n'
    result_str += f'Is valid magic square: {is_valid_magic_square(magic_square)}\n'
    return result_str
 
if __name__ == "__main__":
    orders = list(range(3, 11))
    with Pool() as pool:
        results = pool.map(worker, orders)
    for result in results:
        print(result)

그때 각 선들의 합은 다음 식을 만족합니다.

특이한 것은 마방진 중에서 다음 배열은

각 수를 제곱해도 마방진이 됩니다.

import numpy as np
 
def is_valid_magic_square(magic_square):
    n = len(magic_square)
    magic_sum = np.sum(magic_square, axis=0)[0]
 
    # Check the sum of each row, column, and the two diagonals
    return (
        all(np.sum(magic_square, axis=0) == magic_sum) and
        all(np.sum(magic_square, axis=1) == magic_sum) and
        np.sum(np.diag(magic_square)) == magic_sum and
        np.sum(np.diag(np.fliplr(magic_square))) == magic_sum
    )
 
# Provided magic square
magic_square = np.array([
    [256, 676, 1, 529, 1444, 2704, 1849, 3721],
    [1681, 3969, 1600, 2500, 9, 441, 196, 784],
    [1296, 2916, 2025, 3481, 100, 1024, 49, 289],
    [25, 361, 144, 900, 2209, 3249, 1156, 3136],
    [729, 169, 484, 16, 2401, 1521, 4096, 1764],
    [3844, 1936, 2601, 1369, 576, 4, 625, 225],
    [3025, 1089, 3364, 2304, 841, 121, 400, 36],
    [324, 64, 961, 81, 3600, 2116, 2809, 1225]
])
 
print(is_valid_magic_square(magic_square))

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