•A magic square is an nxn square divided into n2 cells inscribed with n2 disjoint integers arranged in such a way that the integers in every row, every column and the two diagonals sum to the same ...constant which is called the MAGIC CONSTANT of the square.•Magic squares are examples of SameSum graphs. In general, a SameSum graph is a graph whose vertices are inscribed with positive integers which are arranged in such a way that there are several subsets of the inscribed integers which sum to the same constant.•This article introduces the n-trigles, a new variant in the Samesum family.•An n-trigle is a planar connected graph made of triangles. A triplet of distinct positive integers is inscribed at the three vertices of every triangle in the graph. The triplet of integers in every triangle sum to the same constant n. The number of triangles in the graph is determined by a formula that depends on n.
A SameSum graph is a graph whose vertices are inscribed with positive integers which are arranged in such a way that there are several subsets of the inscribed integers which sum to the same constant. An example of a SameSum graph is a Magic Square: the triplets of integers along the rows, along the columns and along the two diagonals of a magic square sum to the same magic constant of that specific square. Some recent results of the author of this article present several groups of SameSum graphs providing some additional examples. This article introduces the n-trigles, a new variant in the SameSum family, as a planar connected graph based on triangles only. The n-trigles are defined, their versatile properties are described and several examples are shown.
Sparse anti-magic squares are useful in constructing vertex-magic labelings for bipartite graphs. An n×n array based on {0,1,…,nd} is called a sparse anti-magic square of ordernwith densityd (d<n), ...denoted by SAMS(n,d), if its row-sums, column-sums and two main diagonal sums constitute a set of 2n+2 consecutive integers. A SAMS(n,d) is called regular if there are d positive entries in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares with densities d=3,5 and it is proved that there exists a regular SAMS(n,3) if and only if n≥4 and there exists a regular SAMS(n,5) if and only if n≥6.
A magic square is an n×n array of numbers whose rows, columns, and the two diagonals sum to μ. A regular magic square satisfies the condition that the entries symmetrically placed with respect to the ...center sum to 2μn. Using circulant matrices we describe a construction of regular classical magic squares that are nonsingular for all odd orders. A similar construction is given that produces regular classical magic squares that are singular for odd composite orders. This paper is an extension of 3.
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