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REVIEW 2 major objections 6 minor 9 references

SET! From Groups to Games

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper argues that SET generalizes to groups in two natural ways—sets as products equal to the identity and sets as arithmetic progressions—and introduces C53T, a five-card game on a Z₅³-torsor whose sets are exactly…

desk verdict Pleasant expository paper with two genuinely new playable games and one cute but under-proved symmetry observation; the alleged counterexample to the order-3 claim doesn't hold up. read the letter →

arxiv 2507.11235 v1 pith:XN4GI5DL submitted 2025-07-15 math.HO

Andrey Boris Khesin , Tanya Khovanova This is my paper · ORCID
classification math.HO MSC 00A0820K0105B99
keywords SETgametorsorcyclicgroupcardgeneralizationarithmeticprogressionpentagonsymmetrynon-abelianrecreationalmathematics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that the classic game SET, usually played with three-feature cards over the group C₃⁴, can be generalized to arbitrary groups in two natural ways: sets as products equal to the identity, and sets as arithmetic progressions. Its central new contribution is the C53T game, played on a torsor of Z₅³, where a five-card set is defined by the five values in each coordinate summing to zero mod 5. The paper claims that such zero-sum five-multisets are exactly the configurations of five marks on a regular pentagon that possess a line of symmetry, and that this visual equivalence is unique to the group sizes 3 and 5. If true, this gives a playable geometric rule for spotting C53T sets and clarifies why the number 5 is special for torsor-based SET variants.

What carries the argument

The load-bearing structures are the torsor and the symmetric-pentagon identity. A torsor is a group whose identity element has deliberately been forgotten, so no card is singled out as special, as in SET. In C53T the relevant identity is that for n = 3 and n = 5, an n-element multiset in Zₙ sums to zero modulo n if and only if the corresponding n marks on a regular n-gon have a reflection symmetry; for n = 5 this turns the algebraic zero-sum condition into a visible axis of symmetry. The arithmetic-progression machinery is the condition ba⁻¹ = cb⁻¹, which generalizes 'b − a = c − b' from vector spaces to arbitrary groups and is what lets non-abelian groups be used as torsors.

What would settle it

Enumerate all multisets of size n in Zₙ for n = 9, 11, 13, and 25 and check whether, for each n, a multiset sums to zero modulo n if and only if its points on a regular n-gon are reflection-symmetric; finding one n > 5 where the two conditions coincide would refute the uniqueness claim, and finding one where they diverge would support it.

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Extended reading notes

Core claim

On the C53T deck, each card carries three pentagons with one marked direction, so a card is an element of a Z₅³-torsor. A set is five cards whose marks, coordinate by coordinate, sum to zero modulo 5. The paper's discovery is that for the cyclic group Z₅, a multiset of five residues sums to zero exactly when those five residues, placed at the vertices of a regular pentagon, admit an axis of reflection symmetry. The same equivalence holds for Z₃, the group underlying original SET, and the paper asserts it fails for every larger group, giving Z₇ as a counterexample. The paper also develops a separate generalization in which three cards a, b, c form a set when ba⁻¹ = cb⁻¹, which makes any group (including non-abelian ones) into a torsor-based SET game and yields the OCTA Set deck on the octahedral group.

Load-bearing premise

The paper's claim that the zero-sum/symmetry equivalence is unique to sizes 3 and 5 rests on only one explicit counterexample (n = 7); if another size n ≥ 6 also satisfies the equivalence, the claimed uniqueness fails.

Editorial extensions

If this is right

  • In C53T, a player can verify a five-card set by checking that each of the three pentagons has an axis of symmetry, without doing modular arithmetic.
  • Because the symmetry equivalence is unique to 3 and 5, no similar visual rule will work for a torsor over Zₙ with n ≥ 6; game designs for larger cyclic groups need a different visual strategy.
  • The arithmetic-progression rule makes every finite group a potential SET game, so the only obstacle is finding a group with a recognizable visualization and few elements of order 2.
  • The OCTA Set construction shows that one card can carry two isomorphic presentations (cube and octahedron) so that different players can solve using whichever structure they see first.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The 'unique to 3 and 5' assertion is supported only by a single Z₇ counterexample; a systematic check of larger n might reveal additional n for which the symmetry equivalence holds, which would change how special 5 is.
  • The pentagon-symmetry rule could be reused for other geometric figures if the underlying group is replaced by a dihedral group where reflection symmetries are built into the object labels, not just the cyclic placement.
  • The arithmetic-progression definition suggests a family of games on groups with no elements of order 2, where every pair of cards completes to a set; C₅ᵏ is one such family and larger k would give larger decks.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper surveys and extends generalizations of the game SET. It considers two families of generalizations: sets defined as tuples of group elements whose product is the identity, and sets defined as arithmetic progressions in a group. The authors introduce the torsor viewpoint, relate ProSet and EvenQuads, propose a new game C53T based on the group C_5^3 (with five-card sets defined by summing to zero in each attribute), and propose OCTA Set based on the octahedral group C_2 ≀ S_3. The final section gives practical design criteria for SET variants. The central mathematical claims are the equivalence, for n=5, between zero-sum multisets of five elements of C_5 and reflection-symmetric configurations on a pentagon, and the assertion that this property is unique to n=3 and n=5.

Significance. If the equivalence for n=5 is proved and the uniqueness claim corrected, the C53T visual rule is a beautiful and pedagogically valuable characterization, and the paper's survey of group-based SET variants is useful to mathematicians and game designers. The new games are concrete, the card designs are clearly described, and the web implementation and references are welcome. However, the uniqueness claim as stated is false for n=3 and unsupported for other n, and another claim about order-3 elements in non-abelian groups is false. These defects are local and repairable, but they currently undermine the paper's most distinctive assertion.

major comments (2)
  1. [Section 3, Figure 5 and following paragraph] The sentence 'This is a property unique to the numbers 3 and 5' is false for n=3. For n=3, the multiset {0,1,2} sums to 0 mod 3 but, placed on an equilateral triangle, has no reflection symmetry; conversely, {0,1,1} is invariant under the reflection fixing the vertex labeled 0 but sums to 2 mod 3. Thus the claimed equivalence 'zero-sum iff reflection-symmetric' fails in both directions for n=3. Moreover, the uniqueness for larger n is not proved: one n=7 example does not exclude other n, and the construction with n−3 zeros plus one each of 1, 2, and n−3 gives a zero-sum multiset with no reflection symmetry for every n≥6. Since the visual rule for C53T is introduced as a theorem, the authors should prove the n=5 classification and either correct the n=3 statement or restrict the claim to n=5.
  2. [Section 4, paragraph beginning 'If ba−1 is an element of order three'] The claim that if ba−1 has order 3 then the completing cards form a set in any order is false in non-abelian groups. In S3, take a=(12) and b=(13); then ba−1=(123) has order 3 and c=ba−1b=(23), but the ordered triple (a,c,b)=((12),(23),(13)) is not an arithmetic progression because (23)(12)−1(23)=(132)≠(13). The statement is true in abelian groups with all nonzero elements of order 3 and in particular in SET, so it should be restricted accordingly.
minor comments (6)
  1. [Section 3, Figure 5 caption] The caption says 'up to rotations', but the equivalence being illustrated is with reflection symmetries; please clarify whether the figure shows representatives up to the full dihedral group or up to rotations only.
  2. [Section 3] The phrase 'This is a property unique to the numbers 3 and 5' should be revised after the mathematical correction; if the intended statement is for n=5 only, say so.
  3. [Section 4] In the sentence 'For any group, our condition is equivalent to ba−1 = cb−1', the multiplicative notation and the order-sensitivity should be made explicit, e.g., by saying that the group operation is written multiplicatively and that the order of the factors is part of the definition.
  4. [Section 4] The phrase 'It can be viewed as Z3 4' is a typesetting error: it should be Z_4^3, and the sentence should clarify that this is a different group structure from the Z_2^6 used for EvenQuads in Section 3.
  5. [References] Reference [8] (OEIS) is listed but never cited in the text; either cite it where relevant or remove it.
  6. [Section 5] In the paragraph on adding cards when no sets exist, the statement 'if only one card is added at a time, then any new set is guaranteed to contain the newly added card' could be phrased more explicitly: any set not containing the new card would have been a set among the cards already on the table.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the C53T pentagon-symmetry rule is a derived observation rather than an assumed input, and the paper's self-citations are not load-bearing.

full rationale

The paper's derivations are self-contained group theory. Section 2 shows that the SET rule is equivalent to summing to zero over F3^4; Section 3 generalizes this to zero-sum multisets in other groups and observes, rather than assumes, that for C5 the zero-sum condition corresponds to reflection symmetry of a pentagon labeling. The 'unique to 3 and 5' assertion is supported only by n=4 and n=7 checks and therefore is under-proved, but an omitted proof is a completeness/correctness issue, not circularity: the equivalence is not defined into existence by the symmetry criterion, nor is it fitted to data. The only self-citation, [1], supports a side remark on the ProSet/EvenQuads correspondence and is not load-bearing for the new C53T, arithmetic-progression, or OCTA Set content. Consequently no step in the claimed derivation chain reduces to its own input.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The new contributions use standard group theory and an unproved symmetry characterization. No free parameters are fit; the only non-standard input is the C5 symmetry assertion.

assumptions (3)
  • domain assumption The SET deck is modeled by F_3^4, and a set is three vectors summing to zero.
    Section 2 establishes this as the starting point; it is the standard mathematical model of SET.
  • standard math The arithmetic progression condition b-a=c-b generalizes to ba^{-1}=cb^{-1} in any group.
    Section 4 derives this by multiplying on the left; it is a basic group-theoretic equivalence.
  • ad hoc to paper Every zero-sum multiset of five elements of Z5 is invariant under some reflection of the pentagon, and this property is unique to n=3 and n=5.
    Stated in Section 3 with a figure and a single counterexample for n=7; no proof is supplied, so the paper's 'special feature' claim rests on this unproved assertion.

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Cite this review

Pith. "Pith review of SET! From Groups to Games." pith.science (2026). https://pith.science/paper/XN4GI5DL

@misc{pith2026250711235,
  author       = {Pith},
  title        = {Pith review of: SET! From Groups to Games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XN4GI5DL}},
  note         = {Machine review of arXiv:2507.11235}
}
read the original abstract

The game of SET is one of the best mathematical games ever. It is no wonder that people have tried to generalize it. We discuss existing generalizations of the game of SET to different groups. We concentrate on two types of generalization: a) where a set consists of cards that multiply to the identity; b) where a set consists of three cards that form an arithmetic progression. We finish with a discussion of some properties of the games that influence how enjoyable they are.

Figures

Figures reproduced from arXiv: 2507.11235 by the authors.

Figure 1
Figure 1. An example of a SET. The underlying group is [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. An example set in a game where the four shown group elements multiply to the identity. This [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. An example set in the game of ProSet, or Socks. Every color of sock appears an even number [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: A quad in EvenQuads. Can we extend the notion of a torsor to ProSet? The answer is yes! EvenQuads is almost equivalent to the game of ProSet [1]. To make ProSet have a torsor-like structure, we just need to add an identity card, only allow sets with four cards, and the…
Figure 5
Figure 5. Figure 5: Consider all possible ways to choose five values such that their sum is 0 mod 5. This figure [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: An example of a set in C53T. The first dimension (light gray pentagons) has a vertical axis [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: A set forming an arithmetic progression using C53T cards. The three cards can be interpreted [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: Two cards, a and b, showing elements of C2 ≀ S3. There is a unique third card that completes an arithmetic set with these two. The first card, a, is its own inverse, so the third card we are looking for is equal to the product bab, which we can visualize as follows. If…
Figure 10
Figure 10. Figure 10: A set in OCTA Set. There are several ways to interpret each card. We can imagine an [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages

  1. [1]

    Card Games Unveiled: Exploring the Underlying Linear Algebra

    Nikhil Byrapuram, Hwiseo (Irene) Choi, Adam Ge, Selena Ge, Tanya Khovanova, Sylvia Zia Lee, Evin Liang, Rajarshi Mandal, Aika Oki, Daniel Wu, and Michael Yang, Card Games Unveiled: Exploring the Underlying Linear Algebra. math.HO arXiv:2306.09280, (2023)

    work page Pith review arXiv 2023

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    How many cards should you lay out in a game of EvenQuads?: A detailed study of caps in AG(n,2)

    Julia Crager, Felicia Flores, Timothy E. Goldberg, Lauren L. Rose, Daniel Rose-Levine, Darrion Thorn- burgh, and Raphael Walker, How many cards should you lay out in a game of EvenQuads? A study of 2-caps in AG(2, n), math.CO arXiv:2212.05353, 2022

    work page Pith review arXiv 2022

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    Grochow, New applications of the polynomial method: The cap set conjecture and beyond, Bulletin of the American Mathematical Society

    Joshua A. Grochow, New applications of the polynomial method: The cap set conjecture and beyond, Bulletin of the American Mathematical Society. 56(1), (2018)

    work page 2018

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    The Joy of SET

    Liz McMahon, Gary Gordon, Hannah Gordon, and Rebecca Gordon. The Joy of SET. The Many Math- ematical Dimensions of a Seemingly Simple Card Game. Princeton University Press, 2017

    work page 2017

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    The Game of Set (and some variations)

    Numberphile video “The Game of Set (and some variations)”, available at https://www.youtube.com/ watch?v=EkFX9jUJPKk

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    Illustrating Mathematics, available at https://im.icerm.brown.edu/portfolio/nonabelian-set/

    Cathy Hsu, Jonah Ostroff, Lucas Van Meter, Gabriel Dorfsman-Hopkins, Nonabelian SET. Illustrating Mathematics, available at https://im.icerm.brown.edu/portfolio/nonabelian-set/

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    Rose, Quads: A SET-like game with a Twist, available at http://sigmaa.maa.org/mcst/ QUADS%20-%20SET%20WITH%20A%20TWIST.pdf

    Lauren L. Rose, Quads: A SET-like game with a Twist, available at http://sigmaa.maa.org/mcst/ QUADS%20-%20SET%20WITH%20A%20TWIST.pdf

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    (2023), The On-Line Encyclopedia of Integer Sequences, Published electronically at https://oeis.org

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    2023

Show all 9 references
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    tsetse website available at https://tsetse.tck.mn/help.html, maintained by Andy Tockman. 10

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