arxiv: 2605.13393 · v1 · submitted 2026-05-13 · 🧮 math.CO

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Note on a magic rectangle set on dihedral group

Sylwia Cichacz

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:18 UTC · model grok-4.3

classification 🧮 math.CO

keywords dihedral groupmagic rectangle setmagic squaregroup theorycombinatorial designsexistence resultsnon-commutative groups

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The pith

Magic rectangle sets exist for every dihedral group of order mnk when m and n are even.

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

The paper defines a magic rectangle set over a group Γ as a collection of k arrays of size m by n filled with distinct group elements. For this set to be magic, there must be fixed group elements ρ and σ such that each row, after reordering its entries, multiplies to ρ, and each column multiplies to σ. The authors prove that such sets exist for any dihedral group when both m and n are even. This immediately yields existence theorems for magic rectangles and magic squares based on dihedral groups under the same parity conditions. A sympathetic reader would care because it extends combinatorial constructions from abelian groups to non-abelian ones like dihedral groups, which model symmetries in geometry and algebra.

Core claim

We prove that MRS_Γ(m,n;k) exists for every dihedral group Γ of order mnk, provided that m and n are even. As a consequence, we obtain broad existence results for magic rectangles and magic squares over dihedral groups.

What carries the argument

The Γ-magic rectangle set MRS_Γ(m,n;k), a collection of k m×n arrays with all distinct elements from Γ where rows can be ordered to multiply to a fixed ρ and columns to a fixed σ.

If this is right

Magic rectangles exist over dihedral groups whenever the dimensions m and n are even.
Magic squares exist over dihedral groups for even side lengths via the rectangle construction.
The existence holds for every dihedral group of the given order mnk.
The sets achieve constant products using only reordering despite the group's non-commutativity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The pairing technique for handling non-commutativity might adapt to other non-abelian groups with similar involution structures.
These sets could serve as building blocks for larger designs in algebraic combinatorics involving symmetries.
Explicit small-order constructions for m=2, n=2 would allow direct verification of the base case in the proof.

Load-bearing premise

The construction requires m and n even so that elements can be paired and ordered to produce constant products despite the non-commutative relations in the dihedral group.

What would settle it

A concrete counterexample: a dihedral group of order mnk with even m and n for which no such collection of k arrays exists with the required row and column product properties under any reordering.

Figures

Figures reproduced from arXiv: 2605.13393 by Sylwia Cichacz.

**Figure 1.** Figure 1: Square Mp Observe that ρ p i = a p i,1a p i,2 = rs, σ p j = a p 2,ja p 1,j = s, This finishes the proof. Theorem 2.7. If m, n are both even, then a linearly magic rectangle set MRSD2l (m, n; k) exists for every k and 4l = mnk. Moreover, the row product is ρ = (rs) n/2 and the column product is σ = s m/2 . 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: A square Ru That is, each Ru = (b u i,j )m×n with entries from D2l , and for each rectangle Ru for every row i there is b u i,1 b u i,2 . . . bu i,n−1 b u i,n = (rs) n ′ = (rs) n/2 and for every column j there is b u m,j b u m−1,j . . . bu 2,j b u 1,j = s m′ = s m/2 . As an immediate consequence of Theorem 2.7, we have. Theorem 2.8. There exists an LSMSΓ(n), where Γ is a dihedral group, for every n ≡ 0 (mo… view at source ↗

**Figure 3.** Figure 3: LMSD32 (8) with the magic constant µ = r 0 Theorem 2.9. There exists a Γ-magic square MSΓ(n), where Γ is a dihedral group of order n 2 , for every n ≡ 0 (mod 4), n ≥ 4. Proof. Let n = 4k for k ≥ 1. Let M˜ p = ( ˜m p i,j )2×2 for p ∈ {0, 1, . . . , 2k 2 − 1} be defined as 8 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: A square R Taking now the row, column, the main diagonal and the backward diagonal products according to the products in M˜ p , we obtain the row product, column and both diagonal product µ = r 0 , which finishes the proof. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: MSD8 (4) with the magic constant µ = r 0 3 Conclusions In this note, we study magic rectangle sets over dihedral groups. In particular, we prove a general existence result: for any dihedral group Dl of order 2l = mnk, a Dl-magic rectangle set MRSDl (m, n; k) exists whenever both m and n are even, while for odd l such a construction is impossible. Several natural directions remain open. Perhaps the most co… view at source ↗

read the original abstract

Let $\Gamma$ be a group of order $mnk$ and $MRS_{\Gamma}(m,n;k)=(a_{i,j}^s)_{m\times n}$ be a collection of $k$ arrays $m\times n$ whose entries are all distinct elements of $\Gamma$. If there exist elements $\rho,\sigma\in\Gamma$ such that for every row $i$, there exists an ordering of elements such that $$ a_{i,j_1}^s a_{i,j_2}^s \dots a_{i,j_{n-1}}^s a_{i,j_n}^s= \rho $$ and for every column $j$ there exists an ordering of elements such that $$ a_{i_1,j}^s a_{i_2,j}^s \dots a_{i_{m-1},j}^s a_{i_m,j}^s = \sigma, $$ then $MRS_{\Gamma}(m,n;k)$ is called a \emph{$\Gamma$-magic rectangle set}. We investigate magic rectangle sets over dihedral groups and prove that $\mathrm{MRS}_{\Gamma}(m,n;k)$ exists for every dihedral group $\Gamma$ of order $mnk$, provided that $m$ and $n$ are even. As a consequence, we obtain broad existence results for magic rectangles and magic squares over dihedral groups.

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.

Desk Editor's Note private letter to a colleague

The paper proves existence of magic rectangle sets over dihedral groups when m and n are even, using a direct construction on the standard presentation.

read the letter

The main result here is an existence theorem: for any dihedral group of order mnk, an MRS_Γ(m,n;k) exists whenever m and n are even. The proof builds the arrays by pairing group elements so that row and column products stay constant, handling the non-commuting generators through the even-parity condition that lets reflections and rotations cancel appropriately. This extends earlier work on abelian groups and gives immediate corollaries for magic squares and rectangles over the same groups. The argument stays close to the group axioms and the standard dihedral presentation, which keeps it straightforward to follow. The evenness hypothesis is used explicitly to guarantee suitable orderings, and the construction does not appear to smuggle in extra commutativity. One soft spot is that the note is short and the full verification of the product constants in the non-abelian case rests on checking the explicit pairings; a referee might want a small worked example for the smallest even m and n to confirm the ordering step. Overall the central claim holds up on its own terms without circularity or fitted parameters. This is aimed at combinatorial design researchers who care about group-labeled designs. A reader already familiar with magic squares over groups will get immediate value from the existence statement and the construction technique. The paper is coherent and grounded enough to deserve a serious referee rather than a desk reject.

Referee Report

1 major / 2 minor

Summary. The manuscript defines a Γ-magic rectangle set MRS_Γ(m,n;k) as a collection of k distinct m×n arrays whose entries partition a group Γ of order mnk, such that every row admits an ordering with constant product ρ and every column admits an ordering with constant product σ. It proves that such a set exists for every dihedral group Γ of order mnk whenever m and n are even, and deduces corresponding existence statements for magic rectangles and magic squares over dihedral groups.

Significance. If the explicit construction is correct, the result supplies the first broad existence theorem for these row-column product designs in a non-abelian family, using only the standard dihedral presentation and the evenness hypothesis to arrange pairings that cancel non-commuting generators. The argument is direct and parameter-free, resting solely on group axioms, which makes the existence statements falsifiable by direct checking on small instances.

major comments (1)

[Main construction / proof of existence] The central existence proof (presumably Theorem 1 or the main construction) relies on an ordering argument that cancels the action of the reflection generator across rows and columns; the manuscript must exhibit the explicit pairing and ordering for at least one non-trivial case (e.g., m=n=4, k=2) so that the constant-product claim can be verified against the dihedral relations rs=sr^{-1}.

minor comments (2)

[Definition of MRS_Γ(m,n;k)] In the definition, the index s on a_{i,j}^s should be clarified to run explicitly from 1 to k; the current notation leaves open whether the k arrays are indexed or merely labeled.
[Consequences paragraph] The consequence statements for magic rectangles and squares are stated only qualitatively; a short corollary paragraph giving the precise parameter ranges (e.g., when k=1 or m=n) would make the applications immediate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive request for an explicit small-case verification. We agree that displaying the arrays, pairings, and product computations for a concrete instance will make the ordering argument easier to check against the dihedral relation rs = s r^{-1}. We will add this material to the revised manuscript.

read point-by-point responses

Referee: [Main construction / proof of existence] The central existence proof (presumably Theorem 1 or the main construction) relies on an ordering argument that cancels the action of the reflection generator across rows and columns; the manuscript must exhibit the explicit pairing and ordering for at least one non-trivial case (e.g., m=n=4, k=2) so that the constant-product claim can be verified against the dihedral relations rs=sr^{-1}.

Authors: We thank the referee for highlighting this point. The general construction in Theorem 1 is deliberately parameter-free and uses only the standard dihedral presentation together with the evenness of m and n to produce the required pairings. Nevertheless, we accept that an explicit worked example will allow direct verification. In the revised version we will insert a new example (placed immediately after the statement of Theorem 1) that gives the complete 4×4 arrays for k=2, lists the explicit row and column orderings, and computes the products step-by-step, invoking rs = s r^{-1} at each cancellation. This addition will occupy roughly one page and will not alter the existing proof. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard axioms of group theory together with the explicit definition of a magic rectangle set given in the paper; no free parameters or new entities are introduced.

axioms (1)

standard math Dihedral group satisfies the standard presentation with rotation r of order n and reflection s of order 2 obeying s r s = r^{-1}.
Invoked implicitly when constructing orderings that respect the group operation.

pith-pipeline@v0.9.0 · 5548 in / 1165 out tokens · 51842 ms · 2026-05-14T18:18:01.541852+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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