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arxiv: 1906.11342 · v1 · pith:2PXP6RLTnew · submitted 2019-06-26 · 🧮 math.CO · math.NT

Magic Polygons and Degenerated Magic Polygons: Characterization and Properties

Danniel Dias Augusto , Josimar da Silva Rocha This is my paper

Pith reviewed 2026-05-25 15:13 UTC · model grok-4.3

classification 🧮 math.CO math.NT
keywords magic polygonsdegenerated magic polygonsmagic sumroot vertexexistence conditionsvertex labelingscombinatorial structuresadditive conditions
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The pith

Magic polygons P(n, k) and degenerated forms D(n, k) have a fixed magic sum and root vertex value determined by the parameters.

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

The paper introduces combinatorial objects called magic polygons P(n, k) and degenerated magic polygons D(n, k). It derives explicit expressions for the common sum that every qualifying line must equal and for the number that must occupy the distinguished root vertex. Existence of valid number assignments is shown to hold only for particular pairs of the parameters n and k. A sympathetic reader would care because the work supplies concrete formulas that turn the additive conditions into determined quantities rather than open searches.

Core claim

In this work we define Magic Polygons P (n, k) and Degenerated Magic Polygons D(n, k) and we obtain their main properties, such as the magic sum and the value corresponding to the root vertex. The existence of magic polygons P (n, k) and degenerated magic polygons D(n, k) are discussed for certain values of n and k.

What carries the argument

The definitions of magic polygon P(n, k) and degenerated magic polygon D(n, k) as n-sided figures whose vertices receive number labels satisfying multiple equal-sum conditions; these definitions carry the argument by converting the sum requirements into fixed values for the magic constant and root vertex.

If this is right

  • Once n and k are fixed the magic sum takes a single determined value.
  • The root vertex is forced to a single specific number.
  • Valid labelings exist only for the pairs (n, k) identified by the existence analysis.
  • The degenerated versions obey parallel formulas under their relaxed geometric conditions.

Where Pith is reading between the lines

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

  • The same vertex-sum approach could be applied to labelings on polyhedra or other graphs with multiple intersecting lines.
  • The forced root value may allow a canonical ordering of all valid labelings by rotation or reflection.
  • Small-n computational enumeration could confirm or refute the existence claims for the smallest parameter pairs.

Load-bearing premise

The combinatorial definitions of P(n,k) and D(n,k) admit consistent number assignments to vertices that satisfy the stated additive conditions for at least some n and k.

What would settle it

An explicit check for a pair (n, k) asserted to admit a magic polygon that produces either inconsistent line sums or no valid distinct-integer labeling at all.

Figures

Figures reproduced from arXiv: 1906.11342 by Danniel Dias Augusto, Josimar da Silva Rocha.

Figure 1
Figure 1. Figure 1: Example of Magic Polygon P(4, 4) 4 5 1 3 2 13 6 10 8 12 11 14 15 16 7 17 9 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example of Magic Polygon P(8, 2) Proof. Let x(t−1)nk+(i−1)k+j be the value correspondig to the point P(t−1)nk+(i−1)k+j , j-th point of the i-th edge of the i-th regular polygon. Each point P(t−1)nk+(i−1)k+j of the magic polygon is labeled for a number x(t−1)nk+(i−1)k+j where t ∈ {1, 2, · · · , k 2 }, i ∈ {1, 2, · · · , n} and j ∈ {1, · · · , k} and the root vertice is 3 [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 3
Figure 3. Figure 3: Example of Magic Polygon P(4, 4) using construction 2.2. The particular case P(n, 2) Although it is a particular case of the general case seen in the previous section, we will use another reasoning for a demonstration of the properties of the magic polygons P(n, 2). A magic polygon P(n, 2) is formed by 2n + 1 points, consisting of the vertices, the midpoints of the edges, and the geometric center of a regu… view at source ↗
Figure 4
Figure 4. Figure 4: Example of Magic Polygon P(4, 2) j ∈ {1, · · · , n}, xj is the value assigned to vertex Vj and yj is the value assigned to midpoint of the side whose ends are the vertices Vj and Vj+1 of the magic polygon, then we obtain a magic polygon with n sides such that, for i ∈ {1, 2, · · · , n 2 − 2}, the values assigned to vertices of the magic polygon satisfy    xi =  i + 1, if i is odd n + i … view at source ↗
Figure 5
Figure 5. Figure 5: Magic Polygon P(4, 2) using construction Proof. If the values assigned to all vertices are odd numbers, then, by definition of Magic Polygon, the values assigned to midpoints are even numbers. This contradicts the fact that the magic sum is an odd number, by Theorems 3 and 4. Corollary 1. There is no Magic Polygon P(n, 2) whose values assigned to all mid￾points are even numbers. Proof. If all values assign… view at source ↗
Figure 6
Figure 6. Figure 6: Example of Degenerated Magic Polygon D(5, 2) Theorem 6. A degenerated magic polygon D(n, k) has the following properties: (i) the magic sum is (k + 1) k 2 (n−2)+k+2 2 ; (ii) the value that corresponds to the root vertex is c = k 2 (n−2)+k+2 2 ; (iii) the sum Sj of the values assigns to the j-th points on the edges in the repre￾sentation of the degenerated magic polygon satisfies Sj = (n − 2)k k 2 (n − 2) +… view at source ↗
Figure 7
Figure 7. Figure 7: Example of Degenerated Magic Polygon D(5, 3) 1 5 4 6 7 2 3 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Example of Degenerated Magic Polygon D(3, 2) Proof. Let P(t−1)k(n−2)+k(i−1)+j be the j-th point of the i-th edge of the t-th largest polygon that represents the degenerated magic polygon D(n, k), considering the clockwise direction, let x(t−1)k(n−2)+k(i−1)+j be the value assigned to the point P(t−1)k(n−2)+k(i−1)+j , xtk(n−2)+1 be the value assigned to the point Ptk(n−2)+1, where j, t ∈ {1, 2, · · · , k} an… view at source ↗
Figure 9
Figure 9. Figure 9: Example of Degenerated Magic Polygon D(5, 2) using construction References [1] W.R. Andress, Basic properties of pandiagonal magic squares, Amer. Math. Monthly 67 (1960) 143–152. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
read the original abstract

In this work we define Magic Polygons P (n, k) and Degenerated Magic Polygons D(n, k) and we obtain their main properties, such as the magic sum and the value corresponding to the root vertex. The existence of magic polygons P (n, k) and degenerated magic polygons D(n, k) are discussed for certain values of n and k.

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

1 major / 0 minor

Summary. The paper defines Magic Polygons P(n, k) and Degenerated Magic Polygons D(n, k), claims to derive their main properties including the magic sum and root-vertex value, and discusses existence for selected parameter pairs (n, k).

Significance. If the claimed closed-form expressions and existence results hold with explicit constructions, the work would introduce new parameterized families of combinatorial labelings that could be of interest in magic-figure and design theory. The approach rests on explicit definitions rather than fitted or self-referential quantities.

major comments (1)
  1. [Abstract] Abstract: the abstract asserts that properties and existence results are obtained, yet supplies no derivations, constructions, or verification steps; the central claims therefore cannot be checked against the paper's own evidence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and the opportunity to respond. The single major comment concerns the abstract. We address it below and note that the manuscript body contains the requested derivations and constructions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the abstract asserts that properties and existence results are obtained, yet supplies no derivations, constructions, or verification steps; the central claims therefore cannot be checked against the paper's own evidence.

    Authors: The abstract is a concise summary by design and does not contain derivations, as is standard. The closed-form expressions for the magic sum and root vertex, together with existence conditions and explicit constructions for selected (n, k), are derived and verified in Sections 2–4 of the manuscript. The claims can therefore be checked directly against the paper's evidence. revision: no

Circularity Check

0 steps flagged

Explicit combinatorial definitions yield derived properties without reduction to inputs or self-citation

full rationale

The paper introduces fresh definitions for P(n,k) and D(n,k) as combinatorial objects on polygons with additive labeling constraints, then derives closed-form expressions for the magic sum and root-vertex value directly from those definitions. Existence statements for selected parameter pairs are likewise obtained by checking consistency against the same additive conditions. No fitted parameters are renamed as predictions, no self-citations supply load-bearing uniqueness theorems, and no ansatz is smuggled in; the derivation chain is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so no concrete free parameters, background axioms, or invented entities can be extracted. The contribution consists of the definitions themselves.

pith-pipeline@v0.9.0 · 5580 in / 1072 out tokens · 35733 ms · 2026-05-25T15:13:34.044291+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

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