Note on zero-sum magic squares on Abelian groups
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918486358360064 |
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| author | Cichacz, Sylwia Froncek, Dalibor |
| author_facet | Cichacz, Sylwia Froncek, Dalibor |
| contents | Let $(Γ,+)$ be an Abelian group of order $n^2$. A $Γ$-magic square of order $n$ is an $n\times n$ array whose entries are pairwise distinct elements of $Γ$ such that all row sums, column sums, and the two main diagonal sums are equal to the same element $μ\in Γ$, called the magic constant.
A combinatorial design is called $Γ$-additive if its point set is a subset of an Abelian group $Γ$ and every block has sum zero. If the point set coincides with $Γ$, the design is said to be strictly $Γ$-additive. Motivated by this notion, we construct $Γ$-magic squares with magic constant $μ=0$ whose rows, columns, and two main diagonals can be used as blocks of a strictly $Γ$-additive design. We call such a square zero-sum $Γ$-magic square.
In this paper, we establish necessary and sufficient conditions for the existence of zero-sum $Γ$-magic squares. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05181 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Note on zero-sum magic squares on Abelian groups Cichacz, Sylwia Froncek, Dalibor Combinatorics Let $(Γ,+)$ be an Abelian group of order $n^2$. A $Γ$-magic square of order $n$ is an $n\times n$ array whose entries are pairwise distinct elements of $Γ$ such that all row sums, column sums, and the two main diagonal sums are equal to the same element $μ\in Γ$, called the magic constant. A combinatorial design is called $Γ$-additive if its point set is a subset of an Abelian group $Γ$ and every block has sum zero. If the point set coincides with $Γ$, the design is said to be strictly $Γ$-additive. Motivated by this notion, we construct $Γ$-magic squares with magic constant $μ=0$ whose rows, columns, and two main diagonals can be used as blocks of a strictly $Γ$-additive design. We call such a square zero-sum $Γ$-magic square. In this paper, we establish necessary and sufficient conditions for the existence of zero-sum $Γ$-magic squares. |
| title | Note on zero-sum magic squares on Abelian groups |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.05181 |