Semi-magic dihedral squares
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917293196312576 |
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| author | Cichacz, Sylwia Froncek, Dalibor |
| author_facet | Cichacz, Sylwia Froncek, Dalibor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20774 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Semi-magic dihedral squares Cichacz, Sylwia Froncek, Dalibor Combinatorics 05B15 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. |
| title | Semi-magic dihedral squares |
| topic | Combinatorics 05B15 |
| url | https://arxiv.org/abs/2602.20774 |