The present work concerns the algebra of semi-magic square matrices. These can be decomposed into matrices of specific rotational symmetry types, where the square of a matrix of pure type always has ...a particular type. We examine the converse problem of categorising the square roots of such matrices, observing that roots of either type occur, but only one type is generated by the functional calculus for matrices. Some explicit construction methods are given. Moreover, we take an observation by N.J. Higham as a motivation for determining bounds on the operator p-norms of semi-magic square matrices.
An associative magic square is a magic square such that the sum of any 2 cells at symmetric positions with respect to the center is constant. The total number of associative magic squares of order 7 ...is enormous and thus, it is not realistic to obtain the number by simple backtracking. As a recent result, Ripatti reported the number of semi-magic squares of order 6 (the magic squares of 6 × 6 without diagonal sum conditions) in 2018. In this research, with reference to Ripatti's method of enumerating semi-magic squares, we have calculated the total number of associative magic squares of order 7. There are exactly 1,125,154,039,419,854,784 associative magic squares of order 7 excluding symmetric patterns.
How rare are magic squares? So far, the exact number of magic squares of order n is only known for n ≤ 5. For larger squares, we need statistical approaches for estimating the number. For this ...purpose, we formulated the problem as a combinatorial optimization problem and applied the Multicanonical Monte Carlo method (MMC), which has been developed in the field of computational statistical physics. Among all the possible arrangements of the numbers 1; 2, …, n(2) in an n × n square, the probability of finding a magic square decreases faster than the exponential of n. We estimated the number of magic squares for n ≤ 30. The number of magic squares for n = 30 was estimated to be 6.56(29) × 10(2056) and the corresponding probability is as small as 10(-212). Thus the MMC is effective for counting very rare configurations.
By treating regular (or associative), pandiagonal, and most-perfect (MP) magic squares as matrices, we find a number of interesting properties and relationships. In addition, we introduce a new class ...of quasi-regular (QR) magic squares which includes regular and MP magic squares. These four classes of magic squares are called “special”.
We prove that QR magic squares have signed pairs of eigenvalues just as do regular magic squares according to a well-known theorem of Mattingly. This leads to the fact that odd powers of QR magic squares are magic squares which also can be established directly from the QR condition. Since all pandiagonal magic squares of order 4 are MP, they are QR. Also, we show that all pandiagonal magic squares of order 5 are QR but higher-order ones may or may not be. In addition, we prove that odd powers of MP magic squares are MP. A simple proof is given of the known result that natural (or classic) pandiagonal and regular magic squares of singly-even order do not exist.
We consider the reflection of a regular magic square about its horizontal or vertical centerline and prove that signed pairs of eigenvalues of the reflected square differ from those of the original square by the factor i. A similar result is found for MP magic squares and a subclass of QR magic squares.
The paper begins with mathematical definitions of the special magic squares. Then, a number of useful matrix transformations between them are presented. Next, following a brief summary of the spectral analysis of matrices, the spectra of these special magic squares are considered and the results mentioned above are established. A few numerical examples are presented to illustrate our results.
In this paper we propose a new method, based on R-C similar transformation method, to study classification for the magic squares of order 5. The R-C similar transformation is defined by exchanging ...two rows and related two columns of a magic square. Many new results for classification of the magic squares of order 5 are obtained by the R-C similar transformation method. Relationships between basic forms and R-C similar magic squares are discussed. We also propose a so called GMV (generating magic vector) class set method for classification of magic squares of order 5, presenting 42 categories in total.
A simple parameterization of 3×3 magic squares Trenkler, Götz; Schmidt, Karsten; Trenkler, Dietrich
International journal of mathematical education in science and technology,
1/15/2012, Letnik:
43, Številka:
1
Journal Article
Recenzirano
In this article a new parameterization of magic squares of order three is presented. This parameterization permits an easy computation of their inverses, eigenvalues, eigenvectors and adjoints. Some ...attention is paid to the Luoshu, one of the oldest magic squares.
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