The Classification of Magic SET Squares

克久拉霍的幻方是一个超立方体

Khajuraho's magic square is an hypercube

该研究指出印度克久拉霍寺庙的泛幻方，其对称性群与四维超立方体的对称性群同构，揭示了该古代幻方的隐藏数学特性。

Applications of AI for Magic Squares

In recreational mathematics, a normal magic square is an $n \times n$ square matrix whose entries are distinctly the integers $1 \ldots n^2$, such that each row, column, and major and minor traces sum to one constant $μ$. It has been proven that there are 7,040 fourth order normal magic squares and 2,202,441,792 fifth order normal magic squares, with higher orders unconfirmed. Previous work related to fourth order normal squares has shown that symmetries such as the dihedral group exist and that (under certain conditions) normal magic squares can be categorized into four distinct subsets. With…

Frierson's 1907 Parameterization of Compound Magic Squares Extended to Orders 3^{l}, l=1,2,3,..., with Information Entropy

Frierson used a powerful parameterization of the pattern of the order 3 associative magic square to construct a family of six related order 9 compound (or composite) magic squares, several of them ancient. Stimulated by Bellew's 1997 extension to order 27, we extend these ideas to all orders that are powers of 3, and in addition find simple formulae for the matrix spectra and entropic measures for all those orders. This construction is fractal and we give numerical results to order 243 which show an information entropy measure converging to a constant value of about 1.168.. for the lowest entr…

Digital Era: Magic Squares and 8th May 2010 (08.05.2010)

In this short note we have produced different kind of magic squares using digital letter having only the algorisms: 0, 1, 2, 5, and 8. The interesting fact in considering these five digits is that the day 8th May 2010 also have these ones (08.05.2010). Moreover, the magic squares presented have some interesting properties, such as: they remains the same if we rotate them by 180 degree, or see in the mirror, or see on the other side of the paper, etc. Two palindrome semi-magic squares of order 3x3 are also given. Still, we have considered other dates having four digits.

Quad 方阵

Quad Squares

本文研究 EvenQuads 牌组中的 4×4 方阵，定义并分类 Latin、半幻、幻及强幻 quad 方阵，给出任意规模牌组中各类方阵的计数，并用整数按位异或刻画 quad。

Upside Down Numerical Equation, Bimagic Squares, and the day September 11

In this short note we have given an equation based on the date 11.09.2001 and presented some magic squares. The magic squares presented are of order 3x3, 4x4, 5x5, 9x9, 16x16 and 25x25. While the magic square of higher order 9x9, 16x16 and 25x25 are bimagic. These magic squares are based on the digits, 1-6-9, 0-1-2, 0-1-2-9 and 0-1-2-6-9. The interesting fact in all these magic squares is that if we gave them a rotation of 180 degrees, they remain again the magic squares. In order to have this rotation, we have used the numbers in digital forms. Moreover, the day of submission of this work (20…

平方幻方的一种新变换

A New Transformation of the Magic Square of Squares

本文证明高斯平方算术三元组与毕达哥拉斯三元组呈3对1对应，借此将平方幻方难题解转换为更大的完美高斯平方结构，并反向生成高斯整数中的非平凡近似解。

Euler and magic squares (De quadratis magicis)

Magic squares have always been and are still fascinating for many people, be it only because of their mathematical properties. Their origin is still but certain : we find no magic squares in Greece, and only a 3x3 one in China at the beginning of our era. Most of their development was made in islamic countries. In Europe, Euler wrote two memoirs and numerous pages on magic squares. One of his problems is the famous "officer problem". (In french : Les carres magiques ont toujours fascine la plupart des gens, tant par leur apparente simplicite que par leur etonnante propriete. Leur origine est t…

Magic squares of subtraction of Adam Adamandy Kochański

The problem of the construction of magic squares occupied many mathematicians of the 17th century. The Polish Jesuit and polymath Adam Adamandy Kochański studied this subject too, and in 1686 he published a paper in Acta Eruditorum titled "Considerationes quaedam circa Quadrata et Cubos Magicos". In that paper he proposed a novel type of magic square, where in every row, column and diagonal, if the entries are sorted in decreasing order, the difference between the sum of entries with odd indices and those with even indices is constant. He called them \emph{quadrata subtractionis}, meaning squa…

Powers of doubly-affine integer square matrices with one non-zero eigenvalue

When doubly-affine matrices such as Latin and magic squares with a single non-zero eigenvalue are powered up they become constant matrices after a few steps. The process of compounding squares of orders m and n can then be used to generate an infinite series of such squares of orders mn. The Cayley-Hamilton theorem is used to understand this property, where their characteristic polynomials have just two terms.