The Existence Of Diagonally Ordered Magic Square

Combinatorial design is an important branch of discrete mathematics, which studies the arrangements and properties of things with specific requirements."Lou-Shu" in the legend of "He-Tu Lou-Shu" before four thousands in China, is a combinatorial design-magic square of order3. Magic square has been used in philosophical thinking, intellectual development, art design, scientific enlightenment, aesthetic value, etc. The diagonally ordered magic square is a new topic in magic square. In this thesis, the constructions and existences of diagonally ordered magic squares are investigated.In this thesis, n denotes a positive integer.A magic square M=(mi,j),0≤i, j≤n-1, is an n×n matrix of n2distinct nonnegative integers with the property that the sum of the n numbers in every row, in every column and in each diagonal is the same. The sum is called the magic number, n is called the order of M. A magic square M of order n is called normal if the entries of M are0,1,...n2-1.A magic square M=(mi,j) of order n is called diagonally ordered (DOMS(n) for short) if both main diagonal and back diagonal, when traversed from left to right, have strictly increasing values, i.e.DOMS(n) was introduced in2004by C. Gomes, M. Sellmann, and it was proved that there exists a DOMS(n) for any1≤n≤19except for n＝2. In this thesis, we determine the existence of DOMS(n) for any positive n.The thesis consists of five chapters, and is organized as follows.In Chapter1, history of magic square is introduced, the definitions of magic square, diagonally ordered magic square, frame mutually orthogonal Latin squares, etc. are given, and main results of the thesis are presented.In Chapter2, the existence of DOMS(n) are solved via frame mutually orthogonal Latin squares with filling holes and computer searching when n>20. Thus, the existence of DOMS(n) are completely solved for any n. In Chapter3, strongly idempotent self-orthogonal row Latin magic arrays(SISORLMA (n) for short) is introduced to solve the existence of nonelementary rational DOMS(n). The relationship between SISORLMA(n) and nonelementary rational DOMS(n) is set up. The existence of SISORLMA(n) when n is odd order is solved by using frame self-orthogonal Latin squares with filling holes and computer searching; and the existence of SISORLMA(n) when n is even order is solved by constructions of v�'v+4. Thus, it is proved that there exists a SISORLMA(n) for any n except for n∈{2,3}. Then it is proved that there exists an nonelementary rational DOMS(n) for any n except for n€{2,3}. It is also proved that there exists rational DOMS(n) for any positive integer n except for n=2.In Chapter4, diagonally ordered orthogonal weakly diagonal Latin squares (DOOWDLS (n) for short) and strong symmetric diagonally ordered orthogonal weakly diagonal Latin squares (SSDOOWDLS(n) for short) are introduced, respectively, to solve the existence of elementary DOMS(n) and elementary symmetric DOMS(n). It is proved that if there exists a pair of (SS)DOOWDLS(n), then there exists an elementary (symmetric) DOMS(n), and the product construction of (SS)DOOWDLS(n) is given. It is proved that there exists an SSDOOWDLS(n) for each n=0,1,3(mod4), and there does exist an SSDOOWDLS(n) for each n=2(mod4). Thus, it is proved that there exists an elementary symmetric DOMS(n) for each n=0,1,3(mod4), and it is also proved that there does exist an elementary symmetric DOMS(n) for each n=2(mod4). Finally, it is proved that there exists a DOOWDLS(n) for each n with two exceptions of n€{2,6} and two possible exceptions of n∈{22,26}. It is also proved that there exists an elementary DOMS(n) for each n with two exceptions of n€{2,6} and two possible exceptions of n€{22,26}.In Chapter5, some problems for further research are listed.