On the Constructions of CK(m)-Magic Squares of Singly Even Orders

Chong, Jing Yan (2025) On the Constructions of CK(m)-Magic Squares of Singly Even Orders. Final Year Project (Bachelor), Tunku Abdul Rahman University of Management and Technology.

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Abstract

This project is the continuation of the previous research conducted by Chia and Kok in 2024, (Chia & Kok, 2024). This project focuses on the systematic construction and identifying the number of the matrices A0, A1, A0*, and A1*, which are central components in the construction of CK(m)-magic squares of singly even orders. In the original CK(m)-magic square construction, the magic square is constructed by superimposing a matrix A, formed of four submatrices A0, A1, A2, and A3, with a matrix B, constructed from two smaller matrices B1 and B2. Among these components, only A0 and A1 are constructed manually through combinatorial methods, subject to fixed row and diagonal constraints. In contrast, A2 and A3 are deterministically derived from A0 and A1, respectively, while B1 is a reduced magic square of size (2m+1)×(2m+1), and B2 is its 90-degree clockwise rotation. For constructing ref-symmetric CK(m)-magic squares, symmetry conditions are imposed by filtering A0 and A1 to obtain A0* and A1*, from which A2* and A3* are derived to construct the matrix A*. Likewise, B1 is the reduced magic square of size (2m+1)×(2m+1) after the vertical reflection is applied on bottom rows, and B1* is derived from B1 after vertical reflection is applied on middle row, thereby establishing the structure of the matrix B. Therefore, our main challenges are to generate the matrices A0, A1, A0*, and A1*, and determine how many of them exist for a given number of m. This endeavor employs Cartesian product enumeration and programmatic algorithm to construct these matrices efficiently. The results contribute to the field of combinatorics, particularly in matrix construction under constraints and combinatorial design. Keywords: Magic squares, Ref-symmetric, Self-complementary magic squares, Singly even orders, Brute-force method, Combination, Counting rule, Cartesian product.