Establishing Infinite Methods of Construction of Magic Squares

2021 ◽  
Vol 4 (1) ◽  
pp. 19-22
Author(s):  
Lohans de Oliveira Miranda ◽  
Lossian Barbosa Bacelar Miranda
Keyword(s):  
Magic Squares ◽  
Large Production ◽  
Order Four

Here we have established infinite methods of building doubly even magic squares from doubly even magic squares of n order (n ≥ 20) which are formed by blocks of order four whose sums of elements of lines, columns and diagonals are all equal at 2n2 + 2. Such a characteristic of these special magic squares causes a large production of other magic squares.

On most perfect magic squares of order four

2018 ◽  
Vol 68 (7) ◽  
pp. 1411-1423
Author(s):  
Oskar Maria Baksalary ◽  
Dietrich Trenkler ◽  
Götz Trenkler
Keyword(s):  
Magic Squares ◽  
Order Four

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.

Magic squares of order four

1982 ◽  
Vol 306 (1495) ◽  
pp. 443-532 ◽  
Keyword(s):  
Analytical Methods ◽  
New Method ◽  
Magic Squares ◽  
Magic Square ◽  
All Solutions ◽  
History Of ◽  
Analytical Proof ◽  
Necessary And Sufficient ◽  
Order Four

A brief history of work on the 4 x 4 magic square is presented, with particular reference to Frenicle’s achievement over 300 years ago of establishing 880 as the number of essentially different squares by using the method of exhaustion (not convincingly repeated except by computer in 1976). He also established several central theorems. Our paper confirms the number 880 by a wholly new method of Frenicle quads and ‘part sums’, which leads to the classification of all solutions into, initially, six genera one of which has no members and thence to the enumeration of all possible solutions by analytical methods only. The working leads also to the first analytical proof independent of solutions that 12 and only 12 patterns formed by linking‘complementary’ numbers within a square are necessary and sufficient to describe all solutions - a fact which has been known since 1908, but not hitherto proved. A second method of construction and partial proof) greatly shortened by what has gone before, is also described. This yields a highly symmetrical list of the 880 magic squares. Together the two methods combine to explain many of the special characteristics and otherwise mysterious properties of these fascinating squares. The complete symmetrical list of squares ends the paper.

SHOULD THE PRIVATE SECTOR PROVIDE PUBLIC CAPITAL?

2007 ◽  
Vol 11 (3) ◽  
pp. 318-346
Author(s):  
SANTANU CHATTERJEE
Keyword(s):  
Open Economy ◽  
Public Capital ◽  
Borrowing Constraints ◽  
International Capital ◽  
Standard Assumption ◽  
Production Externalities ◽  
International Capital Markets ◽  
Private Provision ◽  
Large Production ◽  
Government Provision

The choice between private and government provision of a productive public good like infrastructure (public capital) is examined in the context of an endogenously growing open economy. The accumulation of public capital need not require government provision, in contrast to the standard assumption in the literature. Even with an efficient government, the relative costs and benefits of government and private provision depend crucially on the economy's underlying structural conditions and borrowing constraints in international capital markets. Countries with limited substitution possibilities and large production externalities may benefit from governments encouraging private provision of public capital through targeted investment subsidies. By contrast, countries with flexible substitution possibilities and relatively smaller externalities may benefit either from governments directly providing public capital or from regulation of private providers. The transitional dynamics also are shown to depend on the underlying elasticity of substitution and the size of the production externality.

The Saar Steps into Large Production

1955 ◽  
Vol 11 (4) ◽  
pp. 25-27
Author(s):  
Sanford Griffith
Keyword(s):  
Large Production

Behavior of Some Varieties in the Kernel Group to the Attack of the Main Pathogens

2018 ◽  
Vol 12 (2) ◽  
pp. 1-7
Author(s):  
Sînziana Venera Morărița
Keyword(s):  
Chemical Composition ◽  
Climatic Conditions ◽  
Complex Chemical ◽  
Complex Chemical Composition ◽  
Large Production ◽  
Special Quality

Abstract Although relatively recent, peach culture has grown great in our country due to the special quality of the fruit, its very complex chemical composition and the large production that can be obtained without much effort. Peach is a species slightly adapted to our climatic conditions, suffers from winter frost, but can provide productive and long productions of 10-15 years.

Effect of Viscous Dissipation on MHD Williamson Nanofluid Flow in a Porous Medium

2019 ◽  
Vol 8 (3) ◽  
pp. 5795-5802 ◽  
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Numerical Solution ◽  
Viscous Dissipation ◽  
Flow Problem ◽  
Numerical Study ◽  
Steady State Flow ◽  
Nanofluid Flow ◽  
Shooting Technique ◽  
Order Four

The main objective of this paper is to focus on a numerical study of viscous dissipation effect on the steady state flow of MHD Williamson nanofluid. A mathematical modeled which resembles the physical flow problem has been developed. By using an appropriate transformation, we converted the system of dimensional PDEs (nonlinear) into coupled dimensionless ODEs. The numerical solution of these modeled ordinary differential equations (ODEs) is achieved by utilizing shooting technique together with Adams-Bashforth Moulton method of order four. Finally, the results of discussed for different parameters through graphs and tables.

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