Counting Magic Venn Diagrams
Phillip Morris (University of West Georgia)
-For centuries there has been interest in magic figures, like magic squares and cubes, and magic graphs, for example, for mathematical recreation, but also to further mathematical knowledge [1, 2, 5]. A Magic Venn Diagram (MVD) is a magic figure introduced by Dr. Robinson and in our researched we referenced his work titled Magic Venn Diagrams. In a magic Venn diagram specific regions of a Venn diagram are assigned a label. The sum of all labels of a set must be the same for all sets. For details, see the next section. MVDs are of general interest as many magic figures can be considered to be special cases of magic Venn diagrams.
-In order to explore the structure of MVDs and to spur the research on MVDs, it is important to count the number of MVDs for different numbers of sets and regions. To the best of our knowledge, there has been no published work and MVDs, and so far, counting MVDs has been done only by hand. But counting MVDs becomes quickly unfeasible even for Venn Diagrams with only three sets as the number of MVDs increases quickly as the number of considered labels and regions. In this work, we have developed and implemented an algorithm that can determine and count all MVDs of a given scenario of sets, regions, and labels. The algorithm is based on backtracking search. In our experiments we could determine the number of MVDs for many scenarios that have not yet been calculated.