Critical Sets and Unavoidable Sets of Strictly Concentric Magic Squares of Odd Order and Their Application to Prime Strictly Concentric Magic Squares of Order 5

There has been much interest in the mathematical investigation of critical sets and unavoidable sets in Latin Squares, Sudoku, and their applications to practical problems in areas such as agriculture and cryptology. This paper considers the associated structures of Strictly Concentric Magic Squares (SCMSs) and Prime Strictly Concentric Magic Squares (PSCMSs). A framework of formal definitions is given that leads to the definitions of critical sets and unavoidable sets. Minimal critical sets are of interest in Latin Squares, and in this article, the cardinality of minimal critical sets of SCMS is given for all n, n odd. Two families of unavoidable sets are established for SCMS, leading to a complete classification of unavoidable sets of minimum PSCMS of order 5.