An Algebraic Treatment of Magic Squares

1973 ◽  
Vol 66 (2) ◽  
pp. 101-107
Author(s):  
Ralph Munger
Keyword(s):  
Junior High ◽  
Magic Squares ◽  
Magic Square ◽  
Algebraic Treatment ◽  
Algebraic Activities

Magic squares have always served as a good motivation device for drill in elementary and junior high mathematics. This article will consider an algebraic treatment of the 3×3 magic square to show some algebraic activities that can be carried out by the beginning or the advanced algebra student.

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.

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.

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.

The Versatile Magic Square

2003 ◽  
Vol 8 (5) ◽  
pp. 252-255
Author(s):  
Gale A. Watson
Keyword(s):  
Special Needs ◽  
Elementary Grades ◽  
Magic Squares ◽  
Magic Square

Article demonstrates the transformations that are possible to construct a variety of magic squares, including modifications to challenge students from elementary grades through algebra. An example of using magic squares with students who have special needs is also shared.

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

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Thitarie Rungratgasame ◽  
Pattharapham Amornpornthum ◽  
Phuwanat Boonmee ◽  
Busrun Cheko ◽  
Nattaphon Fuangfung
Keyword(s):  
Linear Algebra ◽  
Standard Addition ◽  
Scalar Multiplication ◽  
Vector Spaces ◽  
Magic Squares ◽  
Magic Square ◽  
Definition Of

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.

scholarly journals Cryptosystems using an improving hiding technique based on latin square and magic square

2020 ◽  
Vol 20 (1) ◽  
pp. 510
Author(s):  
Sahab Dheyaa Mohammed ◽  
Taha Mohammed Hasan
Keyword(s):  
Finite Field ◽  
Latin Square ◽  
Sensitive Data ◽  
Avalanche Effect ◽  
Magic Squares ◽  
P Values ◽  
Encrypted Data ◽  
Magic Square ◽  
Encryption Method ◽  
Randomness Tests

<p>Hackers should be prevented from disclosing sensitive data when sent from one device to another over the network. Therefore, the proposed method was established to prevent the attackers from exploiting the vulnerabilities of the redundancy in the ciphertext and enhances the substitution and permutation operations of the encryption process .the solution was performed by eliminates these duplicates by hiding the ciphertext into a submatrix 4 x4 that chooses randomly from magic square 16x16 in each ciphering process. Two techniques of encrypted and hiding were executed in the encryption stage by using a magic square size 3 × 3   and Latin square size 3 × 3 to providing more permutation and also to ensure an inverse matrix of decryption operation be available. In the hiding stage, the ciphertext was hidden into a 16×16 matrix that includes 16 sub-magic squares to eliminate the duplicates in the ciphertext. Where all elements that uses were polynomial numbers of a finite field of degree Galois Fields GF ( ).  The proposed technique is robust against disclosing the repetition encrypted data based on the result of Avalanche Effect in an accepted ratio (62%) and the results of the output of the proposed encryption method have acceptable randomness based on the results of the p-values (0.629515) of the National Institute of Standards and Technology (NIST) randomness tests. The work can be considered significant in the field of encrypting databases because the repetition of encrypted data inside databases is considered an important vulnerability that helps to guess the plaintext from the encrypted text.</p>

scholarly journals On Multiplicative Magic Squares

10.37236/457 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Javier Cilleruelo ◽  
Florian Luca
Keyword(s):  
Lower Bound ◽  
Minimal Element ◽  
Magic Squares ◽  
Magic Square ◽  
Positive Integers

In this note, we give a lower bound for the distance between the maximal and minimal element in a multiplicative magic square of dimension $r$ whose entries are distinct positive integers.

Sign in / Sign up

Export Citation Format

Share Document