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arxiv: 2605.05181 · v1 · submitted 2026-05-06 · 🧮 math.CO

Note on zero-sum magic squares on Abelian groups

Sylwia Cichacz , Dalibor Froncek This is my paper

Pith reviewed 2026-05-08 16:14 UTC · model grok-4.3

classification 🧮 math.CO
keywords zero-sum magic squaresAbelian groupsadditive designscombinatorial designsmagic squares on groupsgroup sums
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The pith

Necessary and sufficient conditions are established for zero-sum Γ-magic squares on Abelian groups of order n².

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines exactly when zero-sum Γ-magic squares exist for an Abelian group Γ of order n². These squares are n by n arrays that contain each group element exactly once and have every row, every column, and both main diagonals summing to zero. The construction ensures that these lines can serve as the blocks of a strictly Γ-additive design whose point set is the entire group. A reader would care because the result ties the arrangement properties of magic squares directly to the block-sum requirements of additive designs, giving a precise classification of when such objects can be built together.

Core claim

A zero-sum Γ-magic square is an n×n array filled with distinct elements of Γ such that all row sums, column sums, and the two main diagonal sums equal zero, and these lines can be used as blocks of a strictly Γ-additive design. The paper establishes necessary and sufficient conditions for the existence of zero-sum Γ-magic squares.

What carries the argument

The zero-sum Γ-magic square, an n×n filling of distinct group elements in which the n rows, n columns, and two diagonals each sum to the group identity, supplies the blocks for the strictly additive design on the full point set Γ.

Load-bearing premise

The rows, columns, and two main diagonals, each summing to zero, can be used directly as the blocks of a strictly Γ-additive design whose point set coincides with the entire group Γ.

What would settle it

A specific Abelian group Γ of order n² together with an integer n for which the stated conditions hold but no arrangement of the group elements into an n×n array makes all rows, columns, and both diagonals sum to zero.

Figures

Figures reproduced from arXiv: 2605.05181 by Dalibor Froncek, Sylwia Cichacz.

Figure 1
Figure 1. Figure 1: ZMSZ8⊕Z8 (8) Example 3.4. In view at source ↗
Figure 2
Figure 2. Figure 2: ZMSZ2⊕Z8 (4) We now treat two special cases, in which the group Γ contains a small component. Lemma 3.5. Let Γ = Z2 ⊕ Z2 2α−1 and α ≥ 2. Then there exists a ZMSΓ(2α). Proof. We proceed by induction on α. As in the proof of Lemma 3.3 we enlarge a zero-sum square of side 2α to one of side 2α+1 using block decomposition. The base case for α = 2 is shown in view at source ↗
Figure 3
Figure 3. Figure 3: ZMSZ2⊕Z32 (8) Lemma 3.7. Let Γ = H ⊕ Z2 2α−2 for |H| = 4 and α ≥ 2. Then there exists a ZMSΓ(2α). Proof. Observe that H = Z2 ⊕ Z2 or H = Z4. We proceed by induction on α. The base cases for α = 2 are shown in view at source ↗
Figure 4
Figure 4. Figure 4: ZMSZ2⊕Z2⊕Z4 (4) and ZMSZ4⊕Z4 (4) Now let Mα = {(a, b)i,j} 2 α i,j=1 be a ZMSH⊕Z22α−2 (2α), which exists by in￾ductive hypothesis. We build an Mα+1 = ZMSH⊕Z22α (2α+1) out of four sub￾squares Ms,t{(a, b) s,t i,j} 2 α i,j=1, s, t ∈ [1, 2]. The subsquare M1,1 is obtained from Mα by m 1,1 i,j = (a, 4b)i,j for every (a, b)i,j ∈ Mα. Let H = {0, a1, a2, a3}. Without loss of generality, we may assume that 2a1 = 0 a… view at source ↗
Figure 5
Figure 5. Figure 5: ZMSZ2⊕Z2⊕Z16 (8) Theorem 3.9. Let |Γ| = 22α, Γ ∈ G and α ≥ 2. Then there exists a ZMSΓ(2α). Proof. The proof will be by induction on |Γ|. Let α = 2, then the ZMSΓ(4) for Γ ∈ {Z2 ⊕ Z8, Z4 ⊕ Z4, Z2 ⊕ Z2 ⊕ Z4, Z2 ⊕ Z2 ⊕ Z2 ⊕ Z2} are shown in Figures 6–8. Thus assume that α ≥ 3. Let Γ = Z2 β1 ⊕ Z2 β2 ⊕ · · · ⊕ Z2 βt and β1 ≤ β2 ≤ · · · ≤ βt . Since Γ ∈ G, there is t > 1. 7 view at source ↗
Figure 7
Figure 7. Figure 7: ZMSZ2⊕Z2⊕Z4 (4) (0,0,0,0) (0,1,0,0) (0,0,0,1) (0,1,0,1) (1,1,0,0) (1,0,0,0) (1,1,0,1) (1,0,0,1) (0,0,1,0) (0,1,1,0) (0,0,1,1) (0,1,1,1) (1,1,1,0) (1,0,1,0) (1,1,1,1) (1,0,1,1) view at source ↗
Figure 8
Figure 8. Figure 8: ZMSZ2⊕Z2⊕Z2⊕Z2 (4) 3.3 Main result We start with an easy observation. Observation 3.10. Let Γ be an Abelian group of order n 2 . If Γ ∈ G / , then there does not exist a ZMSΓ(n). 8 view at source ↗
Figure 9
Figure 9. Figure 9: An MSZ9 (3) and a ZMSZ9 (3) However, such a translation is not always possible, as shown in view at source ↗
Figure 10
Figure 10. Figure 10: MSZ2⊕Z8 (4) with the magic constant µ = (0, 6) This observation leads to the following natural problem. 9 view at source ↗
read the original abstract

Let $(\Gamma,+)$ be an Abelian group of order $n^2$. A $\Gamma$-magic square of order $n$ is an $n\times n$ array whose entries are pairwise distinct elements of $\Gamma$ such that all row sums, column sums, and the two main diagonal sums are equal to the same element $\mu \in \Gamma$, called the magic constant. A combinatorial design is called $\Gamma$-additive if its point set is a subset of an Abelian group $\Gamma$ and every block has sum zero. If the point set coincides with $\Gamma$, the design is said to be strictly $\Gamma$-additive. Motivated by this notion, we construct $\Gamma$-magic squares with magic constant $\mu=0$ whose rows, columns, and two main diagonals can be used as blocks of a strictly $\Gamma$-additive design. We call such a square zero-sum $\Gamma$-magic square. In this paper, we establish necessary and sufficient conditions for the existence of zero-sum $\Gamma$-magic squares.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper defines a zero-sum Γ-magic square of order n as an n×n array with distinct entries from an Abelian group Γ of order n² such that all row sums, column sums, and both main diagonal sums equal zero. Motivated by the notion of strictly Γ-additive designs (where the point set is Γ and blocks sum to zero), the authors establish necessary and sufficient conditions for the existence of such squares.

Significance. If the claimed characterization holds, the result supplies a direct link between zero-magic-constant squares and additive designs on groups, with the necessity direction following immediately from the fact that the sum of all row sums equals the sum of all group elements. This could support constructions of designs or further study of magic squares in non-cyclic or non-standard groups.

minor comments (2)
  1. The abstract asserts the existence of necessary and sufficient conditions but does not state them explicitly; moving the precise statement (e.g., the condition on the sum of elements of Γ) into the abstract would improve readability.
  2. The motivation section could clarify whether the rows/columns/diagonals automatically form a design by definition or require additional verification that they are distinct blocks.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central result establishes necessary and sufficient conditions for the existence of zero-sum Γ-magic squares, defined as n×n arrays that are bijective onto an Abelian group Γ of order n² with all row, column, and main diagonal sums equal to 0. The claim that these lines form blocks of a strictly Γ-additive design follows immediately by definition from the bijectivity and zero-sum properties, without any additional assumption or reduction. The necessity that the total sum of elements in Γ equals 0 is obtained directly by adding the n row sums (each 0), which is a basic algebraic identity independent of any fitted parameters, self-citations, or prior results by the authors. No load-bearing step reduces to a self-definition, renamed known result, or imported uniqueness theorem; the derivation remains self-contained via direct group-sum arguments and constructions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard definitions of Abelian groups, magic squares, and additive designs. No free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption Γ is an Abelian group of order n²
    This is the explicit setting stated for the definition of a Γ-magic square.
  • domain assumption Entries are pairwise distinct elements of Γ
    Required by the definition of a Γ-magic square.

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Reference graph

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