Magic Squares Research Articles

Generation of all magic squares of order 5 and interesting patterns finding

Abstract This paper presents an enumeration algorithm to generate all magic squares of order 5 based on the ideas of basic form (Schroeppel [7]) and generating vector which is extension of Frénicle Quads (Ollerenshaw and Bondi [6]). The results lead us to extend Frénicle-Amela patterns from the case of order 4 to the case of order 5, which we refer to Frénicle-Amela-Like patterns. We show that these interesting Frénicle-Amela-Like patterns appear simultaneously. The number of these patterns is also calculated.

Vector Spaces of New Special Magic Squares: Reflective Magic Squares, Corner Magic Squares, and Skew-Regular Magic Squares

The definition of a regular magic square motivates us to introduce the new special magic squares, which are reflective magic squares, corner magic squares, and skew-regular magic squares. Combining the concepts of magic squares and linear algebra, we consider a magic square as a matrix and find the dimensions of the vector spaces of these magic squares under the standard addition and scalar multiplication of matrices by using the rank-nullity theorem.

Public Key Cipher with Signature Based on Diffie-Hellman and the Magic Square Problem

In the present paper, wedeveloped a new variant of asymmetric cipher (Public Key) algorithm based on the Discrete Logarithm Problem (DLP) uses Diffie-Hellman key exchange protocol and the mathematical features of magic square. The proposed method exploits the DLP problem in the key exchange by using the final value of Diffie-Hellman key agreement, just as the dimension to the magic construction and through which determines the type of magic square construction if it is (odd, singly-even or doubly-even) magic square as well as through which determines starting number and the difference value. From the other point, it exploits the magic squares problem in encryption/decryption and signing/verifyingoperations. The developed methodextremely speed in the Encryption/Decryption process as well as in the digital signaturethat uses aSecure Hash Algorithm SHA-1, since the proposed cipher does not use logarithm and the factorization as the traditional algorithms in ciphering and deciphering operations just in the mutual exchanging, but it depends mainly upon the magic constant and magic sum as alternate that deduced from the result of multiplied fixed value, which selected randomly and imposesto keep secret, as we shall explained in the next sections.

General Properties of Strongly Magic Squares

General Properties of Strongly Magic Squares

Reverse vertex magic labeling of complete graphs

A reverse vertex magic labeling of a graph G = (V,E) is a bijection such that for every vertex , is a constant . It is well known that all complete graphs Kn admit a reverse magic labeling. In this paper, we present a new proof of this theorem using the concepts of twin factorization and magic square.

Relativistic Quantum Pseudo-telepathy

We analyze the impact of the Unruh eect on quantum Magic Square game. We find the values of acceleration parameter for which quantum strategy yields higher players’ winning probability than classical strategy.

Supplementary notes on Lewis Carroll, Graeco-Latin squares and magic squares with an annexe on Maria Theresa thalers and British banknotes

Following a brief critique of the wording of Lewis Carroll's Alice's Adventures in Wonderland, this article offers some supplementary remarks relating to the solution of a 3 × 3 pseudo-magic square problem posed by Lewis Carroll, a generalisation of the 3 × 3 Lo-Su or luoshu magic square, the problem of installing m2 = 25 or m2 = 36 officers of m different ranks and m different regiments in a m × m Graeco-Latin square (with apposite remarks on the military career of Leonhard Euler's third son and the recent coinage of the Empress Maria Theresa) and a list of fourteen portraits of mathematicians on banknotes paying particular attention to those of Sir Isaac Newton and Florence Nightingale.

An Efficient Algorithm for Constructing all Magic Squares of Order Four

An Efficient Algorithm for Constructing all Magic Squares of Order Four

On the Road Map of Vogel’s Plane

We define “population” of Vogel’s plane as points for which universal character of adjoint representation is regular in the finite plane of its argument. It is shown that they are given exactly by all solutions of seven Diophantine equations of third order on three variables. We find all their solutions: classical series of simple Lie algebras (including an “odd symplectic” one), \({D_{2,1,\lambda}}\) superalgebra, the line of sl(2) algebras, and a number of isolated solutions, including exceptional simple Lie algebras. One of these Diophantine equations, namely \({knm=4k+4n+2m+12,}\) contains all simple Lie algebras, except so\({(2N+1).}\) Among isolated solutions are, besides exceptional simple Lie algebras, so called \({\mathfrak{e}_{7\frac{1}{2}}}\) algebra and also two other similar unidentified objects with positive dimensions. In addition, there are 47 isolated solutions in “unphysical semiplane” with negative dimensions. Isolated solutions mainly belong to the few lines in Vogel plane, including some rows of Freudenthal magic square. Universal dimension formulae have an integer values on all these solutions at least for first three symmetric powers of adjoint representation.

THE EFFECT OF MAGIC-SQUARE GAME IN IMPROVING 10th GRADER STUDENTS’ SPEAKING ACHIEVEMENT

This Study was an experimental research. The objective was to discover thesignificance of using magic square game in improving 10th grader students’speaking achievement. Twenty students were taken as the sample: 10 students forthe experimental group and 10 students for the control group. The students in theexperimental group were given magic square game as the media to practice theirspeaking skill. The magic square game was used to improve students’achievement in speaking, while the control group were taught without the magicsquare game. They only use textbooks. The students’ performance were evaluatedby using Fulcher’s (2003) Foreign Service Institute (FSI) Weighting Table. Thescoring was done by the English teacher and the researcher. The research findingsshows that the reliability of the test was high, i.e. 0,99, whereas the T-observedvalue at the level of significance 0,05 is 20 and it was lower than the T-critic(22).Therefore, the alternative hypothesis was accepted. It can be concluded that thestudents’ speaking achievement taught by using magic square game wassignificantly higher than those whom taught without using magic square game. So,English teacher are suggested to use magic square game in practicing the students’speaking achievement.Key words: Magic-Square game, Speaking Achievement

한국과 뉴질랜드 초등학교 저학년의 덧셈과 뺄셈 지도방법에 관한 고찰

The purpose of this study is to analyze teaching method of addition and subtraction of whole number in Korea and New Zealand lower grade textbook and to get some suggestive points to develop mathematics curriculum and for a qualitative improvement of textbook. To do this, we will analyze focusing on teaching material, type and method of teaching, cases of real teaching and in the case of New Zealand, we will analyze portfolios together to see what kind of things do they deal with related to addition and subtraction. From these analyzing, the results are as follows: First, the guideline of accomplishment of group of year are stated in 2009 revised curriculum in Korea but it is rough. On the other hand, the level of accomplishment from kindergarten to high school are stated divided by eight kinds of thing in New Zealand curriculum. Second, there were common and different points in the aspect of teaching material. The common points are that both of our Korea and New Zealand are using materials related to real life intimately and the diifferent points are to use technology such as calculator and computer. They are more widely used in New Zealand than our Korea. Third, Korea had used routine method mainly but New Zealand had used method to develop creativity of learner such as to write problem corresponding to expression, posing problem corresponding to information, to complete table and find pattern and to write word problem to explain pattern and so on. Fourth, we could see special calculation strategies in the case of teaching addition and subtraction such as concept of double, compensation, various strategy based on counting of number, addition of the same number, magic square, near-double which are not finding in our mathematics textbook. Fifth, in the New Zealand textbook they had used teaching methods inducing curiosity of learner such as finding message and puzzle problem than solving given problem simply.

QUADRADOS MÁGICOS: QUANDO O MÉTODO FALHA

http://dx.doi.org/10.5902/2179460X14642In this article, we discuss the behavior of the solutions of a system of linear congruences related to the uniform step method. This is a method for constructing magic squares of odd order proposed and mathematically analyzed by Lehmer (1929). His analysis has several steps. In the first, involving the discussion of necessary and sufficient conditions for filling the square, Lehmer says that if two numbers keep a certain relationship to each other, they will be designated to occupy the same cell of the square. We conclude here that numbers that fulfill the mentioned relationship never occupy the same cell. In other words, what Lehmer says is not true.

Pseudo Magic Squares

A magic square of order n is an n × n square (matrix) whose entries are distinct nonnegative integers such that the sum of the numbers of any row and column is the same number, the magic constant. In this paper we introduce the concept of pseudo magic squares, i.e., magic squares defined over the ring of integers, without the restriction of distinct numbers. Additionally, we generalize this new concept by introducing a group (ring) structure over it. This new approach can provide useful tools in order to find new non-isomorphic pseudo magic squares.

Improved even order magic square construction algorithms and their applications in multi-user shared electronic accounts

This paper presents improved even order magic square construction algorithms, including both single even order magic square and double even order magic square construction algorithms. Further, in order to show how the algorithms work, two specific magic squares are constructed. Moreover, the correctness of the algorithms is proved, and the complexity analysis of the algorithms is given. Finally, the improved even order magic square construction algorithms are applied in secure communication and authentication areas for multi-user shared electronic account in detail.

A COMBINATORIAL CHARACTERIZATION OF THE LAGRANGIAN GRASSMANNIAN LG(3,6)()

AbstractWe provide a combinatorial characterization of LG(3,6)(${\mathbb{K}}$) using an axiom set which is the natural continuation of the Mazzocca–Melone set we used to characterize Severi varieties over arbitrary fields (Schillewaert and Van Maldeghem, Severi varieties over arbitrary fields,Preprint). This fits within a large project aiming at constructing and characterizing the varieties related to the Freudenthal–Tits magic square.

English

English

Quantum contextual finite geometries from dessins d'enfants

We point out an explicit connection between graphs drawn on compact Riemann surfaces defined over the field [Formula: see text] of algebraic numbers — the so-called Grothendieck's dessins d'enfants — and a wealth of distinguished point-line configurations. These include simplices, cross-polytopes, several notable projective configurations, a number of multipartite graphs and some "exotic" geometries. Among them, remarkably, we find not only those underlying Mermin's magic square and magic pentagram, but also those related to the geometry of two- and three-qubit Pauli groups. Of particular interest is the occurrence of all the three types of slim generalized quadrangles, namely GQ(2, 1), GQ(2, 2) and GQ(2, 4), and a couple of closely related graphs, namely the Schläfli and Clebsch ones. These findings seem to indicate that dessins d'enfants may provide us with a new powerful tool for gaining deeper insight into the nature of finite-dimensional Hilbert spaces and their associated groups, with a special emphasis on contextuality.

Numerous but rare: an exploration of magic squares.

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) × 102056 and the corresponding probability is as small as 10−212. Thus the MMC is effective for counting very rare configurations.

Innovative default prediction approach

This paper introduces a new scoring method for company default prediction. The method is based on a modified magic square (a spider diagram with four perpendicular axes) which is used to evaluate economic performance of a country. The evaluation is quantified by the area of a polygon, whose vertices are points lying on the axes. The axes represent economic indicators having significant importance for an economic performance evaluation. The proposed method deals with magic square limitations; e.g. an axis zero point not placed in the axes origins, and extends its usage for an arbitrary (higher than 3) number of variables. This approach is applied on corporations to evaluate their economic performance and identify the companies suspected to default. In general, a company score reflects their economic performance; it is calculated as a polygon area. The proposed method is based on the identification of the parameters (axes order, parameters weights and angles between axes) needed to achieve maximum possible model performance. The developed method uses company financial ratios from its financial statements (debt ratio, return on costs etc.) and the information about a company default or bankruptcy as primary input data. The method is based on obtaining a maximum value of the Gini (or Kolmogorov–Smirnov) index that reflects the quality of the ordering of companies according to their score values. Defaulted companies should have a lower score than non-defaulted companies. The number of parameter groups (axes order, parameters weights and angles between axes) can be reduced without a negative impact on the model performance. Historical data is used to set up model parameters for the prediction of possible future companies default. In addition, the methodology allows calculating the threshold value of the score to separate the companies that are suspicious to the default from other companies. A threshold value is also necessary for a model true positive rate and true negative rate calculations. Training and validation processes for the developed model were performed on two independent and disjunct datasets. The performance of the proposed method is comparable to other methods such as logistic regression and neural networks. One of the major advantages of the proposed method is a graphical interpretation of a company score in the form of a diagram enabling a simple illustration of individual factor contribution to the total score value.

Luo Shu

Feng Shui, still popularly practiced today, was closely related to philosophy, natural science, geography, environmental science, architecture, metaphysics, and astrology in ancient China. It is basically divided into the Form School and the Compass School. The latter deals with numerology, calculation, orientation, and time. Luo Shu [洛書], associated with the eight trigrams [八卦], being an ancient Chinese magic square, forms the foundation of the Compass School. The original Luo Shu, a 3 × 3 magic square, was not unique in ancient China but the extension of it to a total of 18 to 36 standard charts was unique, which are still used by all Compass School Feng Shui masters. In this article, modern linear algebra, developed only in the mid-19th century, is employed to prove that there is a strong coherence between the 36 charts if they are treated as 36 matrices and such correspondences conscientiously agree with ancient theories of Feng Shui. This article may help to form a scientific base for the systematic understanding, development, and further research of Luo Shu–related applications.