Magic squares: would you believe…?

1974 ◽  
Vol 21 (5) ◽  
pp. 439-441
Author(s):  
David L. Pagni
Keyword(s):  
Main Diagonal ◽  
Magic Squares ◽  
Magic Square ◽  
Square Array ◽  
Magic Constant

A magic square of nth order is a square array of n rows and n columns whose components are n^2 distinct integers. Furthermore, the sum of the numbers in any row, column, or main diagonal must always equal a constant—the “magic constant.– The array in figure I, then, is a magic square of 3rd order whose magic constant is the number 15.

7. Square arrays

2016 ◽  
pp. 109-122
Author(s):  
Robin Wilson
Keyword(s):  
Latin Square ◽  
Latin Squares ◽  
Magic Squares ◽  
Magic Square ◽  
Orthogonal Latin Squares ◽  
Square Array

‘Square arrays’ is concerned with magic squares and latin squares. An n × n magic square, or a magic square of order n, is a square array of numbers (usually the numbers from 1 to n 2) arranged so that the sum of the numbers in each of the n rows, each of the n columns, or each of the two main diagonals is the same. A latin square of order n, is a square array with n symbols arranged so that each symbol appears just once in each row and each column. Orthogonal latin squares are also discussed along with Euler’s 36 officers problem.

scholarly journals Some properties of magSome Properties of Magic Squares of Distinct Squares and Cubesic squares

2019 ◽  
Vol 30 (3) ◽  
pp. 60
Author(s):  
Amal Abdulrahman Ibrahim ◽  
Shatha A. Salman
Keyword(s):  
Positive Integer ◽  
Magic Squares ◽  
Magic Square ◽  
History Of ◽  
Magic Constant

Magic squares is n×n matrix with positive integer entries as well as the sum of rows, columnsand mains diagonal have the same magic constant, one of the most oldest magic square wasdiscovered in china. In this paper the history of magic square is displayed and some definitionof its kind is given the prove of two theorems about properties of magic square is introduced.

Frenicle’s 880 Magic Squares

2017 ◽  
Author(s):  
John Conway ◽  
Simon Norton ◽  
Alex Ryba
Keyword(s):  
Magic Squares ◽  
Magic Square ◽  
Odd Order ◽  
Order Four ◽  
Magic Constant

This chapter discusses magic squares. A magic square of order n is an arrangement of the numbers from 1 to n 2 in an n × n array so that the two diagonals and all the rows and columns have the same sum. This sum is called the magic constant. Bernard Frenicle de Bessy's work on magic squares appears in two papers published in the book Divers ouvrages de mathematique et de physique par Messieurs de l'Academie Royale des Sciences. In his first paper, “Des Quarrez ou Tables Magiques,” Frenicle quotes a rule for constructing magic squares of odd order. However, Frenicle is more famous for his second paper, “Table Generale des Quarrez de Quatres,” in which he enumerates the 880 magic squares of order four. His enumeration has been repeated many times. These later enumerations they have confirmed the remarkable fact that he was correct.

Multiplicative Squares: Magic and Special

1987 ◽  
Vol 80 (1) ◽  
pp. 51-54
Author(s):  
Steven Schwartzman
Keyword(s):  
Magic Squares ◽  
Magic Square ◽  
Generic Term ◽  
Square Array

Much has been written about magic squares. A magic square is a square array of distinct numbers with the property that the sum of the numbers in each row, column, and diagonal is the same. (I will use the generic term line to stand for row, column, or diagonal.) Figure 1 shows a 3 × 3 magic square whose magic sum is 6. (The references cited at the end of this article provide a good introduction to traditional magic squares and some of the uses to which they have been put in the classroom.)

Mathematical RootsL: Magic Squares: Discovering Their History and Their Magic

2001 ◽  
Vol 6 (8) ◽  
pp. 466-470
Author(s):  
Dawn Anderson
Keyword(s):  
Mathematical Knowledge ◽  
Magic Squares ◽  
Magic Square ◽  
Square Array

Magic squares have long been considered a mathematical recreation providing entertainment and an interesting outlet for creating mathematical knowledge. An nth-order magic square is a square array of n2 distinct integers in which the sum of the n numbers in each row, column, and diagonal is the same.

Magic squares of squares over a finite field

2021 ◽  
pp. 111-122
Author(s):  
Stewart Hengeveld ◽  
Giancarlo Labruna ◽  
Aihua Li
Keyword(s):  
Finite Field ◽  
Prime Numbers ◽  
Similar Question ◽  
Magic Squares ◽  
Content Type ◽  
Magic Square ◽  
Twin Primes ◽  
Quadratic Residues ◽  
Perfect Squares ◽  
Open Question

A magic square M M over an integral domain D D is a 3 × 3 3\times 3 matrix with entries from D D such that the elements from each row, column, and diagonal add to the same sum. If all the entries in M M are perfect squares in D D , we call M M a magic square of squares over D D . In 1984, Martin LaBar raised an open question: “Is there a magic square of squares over the ring Z \mathbb {Z} of the integers which has all the nine entries distinct?” We approach to answering a similar question when D D is a finite field. We claim that for any odd prime p p , a magic square over Z p \mathbb Z_p can only hold an odd number of distinct entries. Corresponding to LaBar’s question, we show that there are infinitely many prime numbers p p such that, over Z p \mathbb Z_p , magic squares of squares with nine distinct elements exist. In addition, if p ≡ 1 ( mod 120 ) p\equiv 1\pmod {120} , there exist magic squares of squares over Z p \mathbb Z_p that have exactly 3, 5, 7, or 9 distinct entries respectively. We construct magic squares of squares using triples of consecutive quadratic residues derived from twin primes.

scholarly journals Magic square and Dirac flavor neutrino mass matrix

2020 ◽  
Vol 35 (29) ◽  
pp. 2050183
Author(s):  
Yuta Hyodo ◽  
Teruyuki Kitabayashi
Keyword(s):  
Neutrino Mass ◽  
Mass Matrix ◽  
Neutrino Mass Matrix ◽  
Magic Squares ◽  
Magic Square ◽  
Normal Mass

The magic texture is one of the successful textures of the flavor neutrino mass matrix for the Majorana type neutrinos. The name “magic” is inspired by the nature of the magic square. We estimate the compatibility of the magic square with the Dirac, instead of the Majorana, flavor neutrino mass matrix. It turned out that some parts of the nature of the magic square are appeared approximately in the Dirac flavor neutrino mass matrix and the magic squares prefer the normal mass ordering rather than the inverted mass ordering for the Dirac neutrinos.

scholarly journals Some results on magic squares based on generating magic vectors and R-C similar transformations

2017 ◽  
Vol 5 (1) ◽  
pp. 82-96 ◽  
Author(s):  
Xiaoyang Ma ◽  
Kai-tai Fang ◽  
Yu hui Deng
Keyword(s):  
Transformation Method ◽  
New Method ◽  
Magic Squares ◽  
Magic Square ◽  
Similar Transformation

Abstract 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.

Constructing Magic Square Number Games

1978 ◽  
Vol 26 (2) ◽  
pp. 36-38
Author(s):  
John E. Bernard
Keyword(s):  
Magic Squares ◽  
Educational Value ◽  
Magic Square

How often have you seen children fill page after page with tic-tac-toe games and hoped for a way to direct their energy and enthusiasm into the learning of mathematics? This article describes how magic squares can be used to generate an assortment of number games that are “the same as” tic-tac-toe. Perhaps these games will be prized not only for their educational value, but also because they provide tic-tac-toe with stiff competition in the “interest- getting department.”

The Odd-numbered Magic Square

1989 ◽  
Vol 82 (2) ◽  
pp. 139-141
Author(s):  
James X. Paterno
Keyword(s):  
Elementary School ◽  
Magic Squares ◽  
Magic Square ◽  
Square Configuration

Acommon exercise in elementary school arithmetic is the completion of magic squares, in which the pupil is to enter certain numerals so that the sum of any row, column, or diagonal is constant. The simplest form involves the use of numerals one through n2, where n2 represents the total number of boxes in a square configuration n boxes by n boxes. This article reviews a familar procedure for generating magic squares and points out a surprising pattern.

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