NumPy's magic circle: revealing the black technology of data processing

Definition of magic circle

The magic matrix is ​​an n x n square matrix in which the sum of the numbers in each row, column and diagonal is equal. This constant is called the magic sum. For example, a 3 x 3 magic circle looks like this:

816
357
492

The magic sum of this magic circle is 15, and the sum of the numbers in its rows, columns and diagonals is 15.

Create magic array in NumPy

NumPy provides a convenience function called numpy.mgrid that can be used to create magic circles. This function creates a grid of evenly spaced numbers that can then be manipulated to generate a magic circle.

The following is the code to create a 3 x 3 magic circle using NumPy:

import numpy as np

# 创建一个 3 x 3 的网格，数字范围为 1-9
grid = np.mgrid[1:4, 1:4]

# 计算网格每个位置的距离矩阵
dists = np.abs(grid[0] - grid[1])

# 创建一个布尔掩码，将与对角线相距 1 的位置标记为 True
mask = (dists == 1) | np.eye(3, dtype=bool)

# 根据掩码对网格进行索引，得到魔法阵
magic_square = grid[mask]

Running this code will generate the following magic circle:

816
357
492

Mathematical properties of magic circle

Magic circles have many interesting mathematical properties, including:

• The sum of the numbers in rows, columns and diagonals is equal.
• The magic sum is equal to n(n² 1)/2, where n is the order of the magic circle.
• The magic circle can be flipped or rotated symmetrically in any row, column or diagonal.
• Magic circles can be created using a variety of algorithms, including the De La Loubère, Lucas and Siwo algorithms.

application

Magic arrays are widely used in various fields, including:

• Data processing: Magic array can be used to visualize and analyze multi-dimensional data.
• Computer Graphics: Magic circles can be used to create symmetrical and beautiful graphics.
• Game Theory: Magic circles can be used to design fair games and puzzles.
• Architecture and Design: Magic circles can be used to create structures and patterns with harmonious proportions and symmetry.

in conclusion

NumPy's magic circle function provides powerful tools for data processing and mathematical exploration. It allows developers to easily create square matrices with magical properties, unlocking a wide range of applications and possibilities.

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