Showing papers on "Magic square published in 2000"
Journal Article•
TL;DR: In this paper, the relation between Tit's magic square of Lie algebras and certain Lie algesbras of 3 × 3 and 6 × 6 matrices with entries in alternative algebraes is investigated, by refor- mulating Tit's definition in terms of trialities.
Abstract: This paper is an investigation of the relation between Tit's magic square of Lie algebras and certain Lie algebras of 3 ×3 and 6 × 6 matrices with entries in alternative algebras. By refor- mulating Tit's definition in terms of trialities (a generalisation of derivations), we give a systematic explanation of the symmetry of the magic square. We show that when the columns of the magic square are labelled by the real division algebras and the rows by their split versions, then the rows can be interpreted as analogues of the matrix Lie algebras su(3), sl(3) and sp(6) defined for each di- vision algebra. We also define another magic square based on 2 ×2 and 4 ×4 matrices and prove that it consists of various orthogonal or (in the split case) pseudo-orthogonal Lie algebras.
29 citations
TL;DR: Even Order Regular Magic Squares Are Singular is found to be Singular when the number of squares in an even order is 1, 2, or 3.
Abstract: (2000). Even Order Regular Magic Squares Are Singular. The American Mathematical Monthly: Vol. 107, No. 9, pp. 777-782.
22 citations
TL;DR: In this paper, it is shown that the sum of the numbers in each row, in each column and in each diagonal of an additive magic square of order n is equal to (nj2Xn 1).
Abstract: Only the crustiest of curmudgeons doesn't love magic squares. And even these sour souls know some folklore about magic squares. But how many of us know how many magic squares there are? In [3] Liang-shin Hahn posed the question: How many 4 X 4 multiplicative magic squares are there that consist of the 16 divisors of 1995. In this brief note we answer Hahn's question by establishing a natural bijection between a class of multiplicative magic squares and a class of additive magic squares (cf. [l]). A multiplicative (respectively, additive) magic square is an n X n matrix of integers in which the product (respectively, sum) of the numbers in each row, in each column and in each diagonal is the same. An additive magic square of order n is an n X n additive magic square whose entries consist of the numbers 0, 1, ... , n 1. It is easy to show that the sum of the numbers in each row, in each column and in each diagonal of an additive magic square of order n is equal to (nj2Xn 1). Much is known about additive magic squares [2], while surprisingly little has been written about multiplicative magic squares. We offer the present paper as a modest remedy to this situation. Let c be the product of n distinct prime numbers. Thus, c has 2\" factors. For example, 1995 is the product of four primes: 3, 5, 7, 19, and has sixteen factors: 1, 3, 5, 7, 15, 19, 21, 35, 57, 95, 105, 133,285,399,665, 1995. If n is even, let Me n be the set of all 2\" 1 X 2\" 1 multiplicative magic squares each of whose entries c~nsists of the 2\" factors of c. Let A 11 be the set of all additive magic squares of order n. The following facts are easy to establish:
11 citations
4 citations
TL;DR: Some patterns of refined epitomes of pansystems methodology were revealed roles and the related of them in problem-solving, modeling, algorithm-generating and theory-constructing were introduced as discussed by the authors.
Abstract: Some patterns of refined epitomes of pansystems methodology were revealed roles and the related of them in problem-solving , modeling, algorithm-generating and theory-constructing were introduced. An important application of pansystems methodology is to give some methods of constructing the typical pansymmetries-magic squares. 1 . a method of recursively constructing magic squares of order n ( n ⩾ 5) ; 2. when magic squares of order m ( m ⩾ 3) and magic squares of order n ( n ⩾ 3) are given a formula of obtaining magic squares of order mn ; 3. when magic squares of order m ( m ⩾ 3) are given, a method of obtaining magic squares of order 2 m .
4 citations
TL;DR: In this article, the construction of the Associated Magic Squares of Order (AOMOS) is discussed. But the authors focus on the Lunda-designs and construction of APOS.
Abstract: (2000). On Lunda-designs and the Construction of Associated Magic Squares of Order 4p. The College Mathematics Journal: Vol. 31, No. 3, pp. 182-188.
3 citations
TL;DR: In this paper, the authors describe tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the popular initials are scratched, birthplaces, houses, or memorials, and a map or directions so that others may follow in their tracks.
Abstract: Does your home town have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.
2 citations