### Abstract

Magic Squares as a mathematical structure have existed for 5000 years, yet they are still an interesting topic of new research. This thesis presents general denitions, examples and important properties of Strictly Concentric Magic Squares (SCMS). Using the known minimum centre cell value of Prime Strictly Concentric Magic Squares (PSCMS) of order 5, some structural properties are established, enabling the production of an algorithm for construction of minimum PSCMS. The number of minimum PSCMS of order 5 is enumerated.Partial SCMS are then introduced with important denitions on completability of grids, with relation to known concepts in Latin Squares and Sudoku grids. The cardinality of sets for dierent types of completability are given in general, where possible, for grids of order n.

The idea of unavoidable sets is introduced on SCMS before specic patterns for the minimum PSCMS of order 5 are given. Having focused on PSCMS of order 5, this thesis then investigates the structure in general for PSCMS of higher odd order. Using the known minimum centre cell value of PSCMS of order 7, an algorithm for construction of these grids is given and the number of minimum PSCMS of order 7 is enumerated. PSCMS of even order are discussed brie y with denitions

that dier from the odd order given as well as an algorithm for construction of a PSCMS of order 6.

The concept of water retention is introduced, rstly on Normal Number Squares, then Prime Number Squares before applying the concept to the minimum PSCMS of order 5. Denitions of patterns are given formally as well as a comparison of known results. Maximum water retention is found in specic cases and compared on identied types of minimum PSCMS of order 5.

Finally, this thesis concludes with a discussion of possible future work.

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Date of Award | 28 Apr 2021 |
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Original language | English |

Supervisor | Stephanie Perkins (Supervisor) & Paul Roach (Supervisor) |