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As matrix representations of magic labelings of related hypergraphs, magic squares and their various variants have been applied to many domains. Among various subclasses, trimagic squares have been investigated for over a hundred years. The existence problem of trimagic squares with singly even orders and orders 16n has been solved completely. However, very little is known about the existence of trimagic squares with other even orders, except for only three examples and three families. We constructed normal trimagic squares by using product constructions, row–square magic rectangles, and trimagic pairs of orthogonal diagonal Latin squares. We gave a new product construction: for positive integers p, q, and r having the same parity, other than 1, 2, 3, or 6, if normal p × q and r × q row–square magic rectangles exist, then a normal trimagic square with order pqr exists. As its application, we constructed normal trimagic squares of orders 8q3 and 8pqr for all odd integers q not less than 7 and p, r ∈ 7, 11, 13, 17, 19, 23, 29, 31, 37. Our construction can easily be extended to construct multimagic squares.

Large-scale unmanned aerial vehicle (UAV) formations are vulnerable to disintegration under electromagnetic interference and fire attacks. To address this issue, this work proposed a distributed formation method of UAVs based on the 3 × 3 magic square and the chain rules of visual reference. Enlightened by the biomimetic idea of the plane formation of starling flocks, this method adopts the technical means of airborne vision and a cooperative target. The topological structure of the formation’s visual reference network showed high static stability under the measurement of the network connectivity index. In addition, the dynamic self-healing ability of this network was analyzed. Finally, a simulation of a battlefield using matlab showed that, when the loss of UAVs reaches 85% for formations with different scales, the UAVs breaking formation account for 5.1–6% of the total in the corresponding scale, and those keeping formation account for 54.4–65.7% of the total undestroyed fleets. The formation method designed in this paper can maintain the maximum number of UAVs in formation on the battlefield.

The efficiency of any cryptosystem not only depends on the speed of the encryption and decryption processes but also on its ability to produce different ciphertexts for the same plaintext. RSA, the public key cryptosystem, is the most famous and widely accepted cryptosystem, but it has some security vulnerabilities because it produces the same ciphertext for identical plaintexts occurring in several places. To enhance the security of RSA, magic square-based encoding models have been proposed in the literature. Although magic square-based encoding models have been proposed, they are static. Thus, this paper introduces a dynamic-based magic square with RSA, where encryption and decryption are performed using numbers generated from the magic square instead of ASCII values. Unlike the static magic square, the proposed dynamic magic square allows users to specify the starting and ending numbers in any position rather than fixed positions. In the proposed dynamic magic square generation, different 4 × 4 magic square templates are created, and 16 × 16 magic squares are generated from them. Experimental results clearly demonstrate the improved security of RSA.

Humanity's love affair with mathematics and mysticism reached a critical juncture, legend has it, on the back of a turtle in ancient China. As Clifford Pickover briefly recounts in this enthralling book, the most comprehensive in decades on magic squares, Emperor Yu was supposedly strolling along the Yellow River one day around 2200 B.C. when he spotted the creature: its shell had a series of dots within squares. To Yu's amazement, each row of squares contained fifteen dots, as did the columns and diagonals. When he added any two cells opposite along a line through the center square, like 2 and 8, he always arrived at 10. The turtle, unwitting inspirer of the ''Yu'' square, went on to a life of courtly comfort and fame. Pickover explains why Chinese emperors, Babylonian astrologer-priests, prehistoric cave people in France, and ancient Mayans of the Yucatan were convinced that magic squares--arrays filled with numbers or letters in certain arrangements--held the secret of the universe. Since the dawn of civilization, he writes, humans have invoked such patterns to ward off evil and bring good fortune. Yet who would have guessed that in the twenty-first century, mathematicians would be studying magic squares so immense and in so many dimensions that the objects defy ordinary human contemplation and visualization? Readers are treated to a colorful history of magic squares and similar structures, their construction, and classification along with a remarkable variety of newly discovered objects ranging from ornate inlaid magic cubes to hypercubes. Illustrated examples occur throughout, with some patterns from the author's own experiments. The tesseracts, circles, spheres, and stars that he presents perfectly convey the age-old devotion of the math-minded to this Zenlike quest. Number lovers, puzzle aficionados, and math enthusiasts will treasure this rich and lively encyclopedia of one of the few areas of mathematics where the contributions of even nonspecialists count.

In 1925 Élie Cartan introduced the principal of triality specifically for the Lie groups of type D_4, and in 1935 Ruth Moufang initiated the study of Moufang loops. The observation of the title in 1978 was made by Stephen Doro, who was in turn motivated by the work of George Glauberman from 1968. Here the author makes the statement precise in a categorical context. In fact the most obvious categories of Moufang loops and groups with triality are not equivalent, hence the need for the word \"essentially.\".

A domino antimagic square of order n is an n×n array formed from a subset of the standard set of 28 dominoes such that the sums of the rows, columns, and two main diagonals form a set of 2n + 2 distinct, consecutive integers while an m × n domino antimagic rectangle is an m × n rectangular array formed from a subset of the standard set of 28 dominoes such that the sums of the rows and columns form a set of m + n distinct, consecutive integers. This paper outlines what the possible dimensions are for m×n domino antimagic rectangles and provides many examples of both domino antimagic rectangles and squares. Many open questions are given at the end of the paper for future exploration.

Magic squares have been widely studied, with publications of mathematical interest dating back over 100 years. Most studies construct and analyse specific subsets of magic squares, with some exploring links to puzzles, number theory, and graph theory. The subset of magic squares this paper focuses on are those termed prime strictly concentric magic squares (PSCMS), and their general definitions, examples, and important properties are also presented. Previously, only the minimum centre cell values of PSCMS of odd order 5 to 19 were presented, by Makarova in 2015. In this paper, the corresponding list of primes for all minimum PSCMS of order 5 is given, and the number of minimum PSCMS of order 5 is enumerated.

This article proposes a first approach to the possibilities of adapting mathematical elements derived from chess to musical composition based on the analysis of the composition process of the work Chess Studies for String Quartet, composed in 2020 by the 1st author of this paper. The structure of its four movements arises from the superimposition of the geometry derived from a sequence of moves for the execution of Bishop and Knight checkmate on a magic square of the 8x8 board, a magic square of Al-Zarquali (11th century). The analysis delves into the evolution of the musical material used, showing the tendency to give rise to symmetries of pitches and temporal fragments between the different movements of the composition. These symmetries give rise to analogies with the concept of chirality, characteristic of the geometry of chess. Structures that show chirality appear in the work both in sound and temporal space. Este artigo propõe uma primeira abordagem sobre as possibilidades de adaptação de elementos matemáticos derivados do xadrez para a composição musical a partir da análise da obra Chess Studies para quarteto de cordas, escrita em 2020 pelo 1º autor deste artigo. A estrutura dos seus quatro movimentos surge da sobreposição da geometria derivada de uma sequência de movimentos para a execução do xeque-mate do Bispo e do Cavalo num quadrado mágico do tabuleiro 8x8, neste caso, o quadrado mágico de Al-Zarquali (século XI). A análise aprofunda a evolução do material musical utilizado, mostrando a tendência para dar origem a simetrias de alturas e fragmentos temporais entre os diferentes movimentos da composição. Estas simetrias dão origem a analogias com o conceito de quiralidade, característico da geometria do xadrez. Estruturas que apresentam quiralidade surgem na obra tanto no espaço sonoro como no espaço temporal. Este artículo propone un primer abordaje sobre las posibilidades de adaptación de elementos matemáticos derivados del ajedrez a la composición musical a partir del análisis de la obra Chess studies para cuarteto de cuerda, escrita en 2020 por el 1er autor de este artículo. La escritura de sus cuatro movimientos surge da la superposición de la geometría derivada de una secuencia de movimientos para la ejecución del jaque-mate de Alfil y Caballo sobre un cuadrado mágico de 8x8, en este caso, un cuadrado mágico de Azarquiel (siglo XI). El análisis profundiza sobre el material musical utilizado, mostrando su tendencia a dar origen a simetrías de alturas y fragmentos temporales entre los diferentes movimientos de la composición. Estas simetrías dan origen a analogías con el concepto de quiralidad, característico de la geometría del ajedrez. Estructuras que presentan quiralidad surgen en la obra tanto en el espacio sonoro como en el temporal.

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.