Critical Sets and Unavoidable Sets of Strictly Concentric Magic Squares and Prime Strictly Concentric Magic Squares of Odd Order

The completability of partial grids has previously been investigated in the literature on Latin Squares and Sudoku grids, and this concept is here defined for strictly concentric magic squares (SCMS) and prime strictly concentric magic squares (PSCMS). This paper provides the first formal definitions concerning the completability of partial SCMS and PSCMS, leading to definitions for their critical sets and unavoidable sets. The cardinality of minimal critical sets is given for grids of order $n$, $n$ odd. Two families of unavoidable sets are established on SCMS, before providing a complete classification for unavoidable sets of minimum PSCMS of order 5.