Dice Relabeling Using Square-Sided Dice

Cooper Stone; Evelyn Fiore; George D. Nasr

arxiv: 2606.20311 · v1 · pith:WPB7YFF7new · submitted 2026-06-18 · 🧮 math.CO

Dice Relabeling Using Square-Sided Dice

Evelyn Fiore , George D. Nasr , Cooper Stone This is my paper

Pith reviewed 2026-06-26 16:51 UTC · model grok-4.3

classification 🧮 math.CO

keywords dice relabelingcyclotomic polynomialssquare-sided dicesum distributionperfect squaresprobability preservation

0 comments

The pith

Pairs of dice with distinct perfect square side counts can be relabeled using cyclotomic polynomials to match the sum distribution of two standard dice.

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

The paper extends a technique using cyclotomic polynomials to relabel dice faces so that the distribution of sums remains the same as with two ordinary dice. It focuses on cases where both dice have a number of sides that is a perfect square, but the two squares are different. Explicit constructions are given for such pairs, along with conjectures about which square side counts allow this. A sympathetic reader would care because it shows the relabeling method works even under the added constraint of square side numbers.

Core claim

There exist pairs of dice with distinct perfect square numbers of sides that admit relabelings generated by cyclotomic polynomials such that the sums have the same frequency distribution as two standard six-sided dice.

What carries the argument

Cyclotomic polynomials applied to generate face labelings for dice with square side counts.

If this is right

Constructions exist for specific pairs of square side counts.
The relabeling preserves sum probabilities for these restricted dice.
Conjectures identify patterns or conditions for larger square side numbers to work.
Future work can explore more pairs based on these ideas.

Where Pith is reading between the lines

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

This suggests the cyclotomic method is flexible enough to accommodate square side constraints without breaking.
Such dice could be used in games or teaching where the number of sides has mathematical significance like being squares.
Testing the conjectures could reveal if all pairs of distinct squares work or only some.

Load-bearing premise

The cyclotomic-polynomial construction developed for arbitrary side counts continues to produce valid relabelings when both side counts are perfect squares.

What would settle it

A specific pair of distinct perfect squares for which the cyclotomic construction yields labelings whose sum distribution does not match that of standard dice.

read the original abstract

We continue recent work of Chao, Gabel, Larson, and Nasr in using cyclotomic polynomials for dice relabeling. In their work, one idea they expand on is finding pairs of dice with different number of sides which maintain the sum frequency of two normal dice. We continue this idea in this paper by studying pairs of dice where the number of sides of each is a different perfect square (which we call "square-sided" dice). We additionally provide conjectures offering ideas for future exploration.

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

Short note applying the existing cyclotomic dice method to square side counts, listing some explicit pairs and leaving a few conjectures open.

read the letter

The main takeaway is that this paper takes the cyclotomic construction from Chao, Gabel, Larson, and Nasr and checks what happens when both dice must have distinct perfect square numbers of sides. The authors produce some concrete pairs that work for the standard sum distribution and add a handful of conjectures.

What stands out is the explicit pairs. The general method already existed, but the square restriction had not been examined, so finding working examples in that subclass adds a few new items to the catalog. That is a straightforward but useful extension.

The limitation is that nothing new is developed algebraically. The paper imports the prior conditions wholesale and reports which square pairs satisfy them, without new proofs or analysis of why squares behave as they do. The conjectures are presented as ideas for later work rather than claims with supporting checks.

If the listed pairs have been properly verified for non-negative coefficients and correct sum frequencies, the existence result holds. The contribution is then the verified examples themselves.

This is narrow material aimed at people already tracking dice relabeling or cyclotomic applications in combinatorics. A reader outside that niche will not find broader implications.

I would send it for peer review. The examples are checkable and the square case is a legitimate specialization that had not been done before.

Referee Report

2 major / 0 minor

Summary. The manuscript continues the cyclotomic-polynomial approach to dice relabeling introduced by Chao et al., restricting attention to pairs of dice whose side counts are distinct perfect squares. It studies this restricted case and supplies conjectures for future work rather than explicit constructions or proofs.

Significance. If the conjectures can be resolved with concrete, verifiable relabelings, the work would show that the algebraic conditions of the prior cyclotomic factorization remain satisfiable inside the sublattice of perfect squares, modestly extending the combinatorial reach of the method. At present the contribution is exploratory and the significance is correspondingly modest.

major comments (2)

[Abstract] Abstract: the manuscript states that the authors 'study' the square-sided case and 'provide conjectures' for future exploration; no concrete pairs, coefficient lists, or verification that the Chao et al. factorization produces non-negative integer multiplicities summing to the required number of faces are supplied, leaving the existence claim unverified.
[Main text (conjectures section)] No section supplies an explicit check that the cyclotomic construction, when both n and m are required to be distinct perfect squares, yields a valid relabeling (non-negative coefficients, correct total faces, and matching sum distribution).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their review. The manuscript is deliberately exploratory: it applies the cyclotomic-polynomial method to the restricted setting of distinct perfect-square side counts and formulates conjectures rather than supplying constructions. We respond to the major comments below.

read point-by-point responses

Referee: [Abstract] Abstract: the manuscript states that the authors 'study' the square-sided case and 'provide conjectures' for future exploration; no concrete pairs, coefficient lists, or verification that the Chao et al. factorization produces non-negative integer multiplicities summing to the required number of faces are supplied, leaving the existence claim unverified.

Authors: The abstract accurately describes the paper's scope. The work studies the square-sided restriction and poses conjectures; it does not assert or verify existence of relabelings within the present manuscript. The absence of explicit coefficient lists or numerical checks is therefore consistent with the stated purpose rather than an omission of a claimed result. revision: no
Referee: [Main text (conjectures section)] No section supplies an explicit check that the cyclotomic construction, when both n and m are required to be distinct perfect squares, yields a valid relabeling (non-negative coefficients, correct total faces, and matching sum distribution).

Authors: The conjectures section is intentionally limited to identifying the setting and stating open questions. Performing the requested explicit checks would constitute a resolution of those conjectures, which lies outside the exploratory scope of the note. The manuscript makes no claim that such verifications have been carried out. revision: no

Circularity Check

0 steps flagged

No circularity; paper studies extension and offers conjectures without deriving claims by construction or load-bearing self-citation

full rationale

The manuscript imports the cyclotomic relabeling method from the external Chao et al. reference (overlapping author Nasr) but explicitly frames its contribution as 'studying' the square-sided restriction and 'providing conjectures' for future work rather than asserting or deriving existence via the construction. No equations, fitted parameters, self-definitions, or reductions of the target distribution to inputs appear in the abstract or described content. The central claim therefore does not reduce to its own inputs or to an unverified self-citation chain; the prior method remains external support even if its applicability to squares is untested here.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, axioms, or invented entities can be extracted beyond the background assumption that cyclotomic polynomials from prior literature apply unchanged to square side counts.

pith-pipeline@v0.9.1-grok · 5600 in / 999 out tokens · 24726 ms · 2026-06-26T16:51:47.672304+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 1 linked inside Pith

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[8]

Lewald and A

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Morrison

I. Morrison. Sacks of dice with fair totals.arXiv preprint arXiv:1411.2272, 2014

Pith/arXiv arXiv 2014
[10]

Rosin, M

A. Rosin, M. Sharobiem, and R. Swift. Dice sums.Mathematical Scientist, 33(2):99 – 109, 2008

2008
[11]

Shen and C

Z. Shen and C. M. Marston. A study of a dice problem.Applied mathematics and computation, 73(2-3):231–247, 1995. Department of Mathematics, Augustana University Email address:george.nasr@augie.edu Email address:ljferguson22@ole.augie.edu Email address:castone23@ole.augie.edu

1995

[1] [1]

Bermudez, A

F. Bermudez, A. Medina, A. Rosin, and E. Scott. Are stupid dice necessary?The College Mathematics Journal, 44(4):315–322, 2013

2013

[2] [2]

D. M. Broline. Renumbering of the faces of dice.Mathematics Magazine, 52(5):312–315, 1979

1979

[3] [3]

B. W. Brunson and R. J. Swift. Equally likely sums.Mathematical Spectrum, 30:34–36, 1998

1998

[4] [4]

Y.-S. Chao, J. Gabel, C. Larson, and G. D. Nasr. Revisiting dice relabeling using cyclotomic polynomials. Enumerative Combinatorics and Applications, 2024

2024

[5] [5]

B. C. Fowler and R. J. Swift. Relabeling dice.The College Mathematics Journal, 30(3):204–208, 1999

1999

[6] [6]

J. A. Gallian and D. J. Rusin. Cyclotomic polynomials and nonstandard dice.Discrete Mathematics, 27(3):245– 259, 1979

1979

[7] [7]

M. Gardner. Mathematical games.Scientific American, 222(6):132–140, 1970

1970

[8] [8]

Lewald and A

A. Lewald and A. Rosin. Equally likely dice sums.Mathematical Scientist, 43(2), 2018

2018

[9] [9]

Morrison

I. Morrison. Sacks of dice with fair totals.arXiv preprint arXiv:1411.2272, 2014

Pith/arXiv arXiv 2014

[10] [10]

Rosin, M

A. Rosin, M. Sharobiem, and R. Swift. Dice sums.Mathematical Scientist, 33(2):99 – 109, 2008

2008

[11] [11]

Shen and C

Z. Shen and C. M. Marston. A study of a dice problem.Applied mathematics and computation, 73(2-3):231–247, 1995. Department of Mathematics, Augustana University Email address:george.nasr@augie.edu Email address:ljferguson22@ole.augie.edu Email address:castone23@ole.augie.edu

1995