Semi-magic dihedral squares

Abstract

Let $Γ$ be a group of order $n^2$ and $SMS_Γ(n)=(a_{i,j})_{n\times n}$ be an $n\times n$ array whose entries are all distinct elements of $Γ$. If there exists an element $μ\inΓ$ such that for every row $i$, there exists an ordering of elements such that $$ a_{i,j_1} a_{i,j_2} \dots a_{i,j_{n-1}} a_{i,j_n} = μ $$ and for every column $j$ there exists an ordering of elements such that $$ a_{i_1,j} a_{i_2,j} \dots a_{i_{m-1},j} a_{i_m,j} = μ, $$ then $SMS_Γ(n)$ is called a \emph{$Γ$-semi-magic square of side $n$} and $μ$ is called a \emph{magic constant}. We provide a complete characterization of semi-magic squares of side $n$ whose entries belong to a dihedral group $D_k$. Moreover, we show that in our constructions a single semi-magic square may admit two distinct magic constants, depending on the order in which the products are computed.

Figures (13)

• Figure 1: Example of $E(4)$, $O(4)$ and $T(4)$
• Figure 2: Example of $E(5)$, $O(5)$ and $T(5)$
• Figure 3: Squares $Q^{11}$ and $Q^{22}$
• Figure 4: Squares $Q^{12}$ and $Q^{21}$
• Figure 5: A square $Q=SMS_{D_{2m^2}}(2m)$

Theorems & Definitions (19)

• Theorem 1.1: handbook
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 2.4
• Theorem 2.5: Froncek_dih_SMS_n=0mod4
• Definition 3.1
• Example 3.3
• Example 3.4
• Theorem 4.3