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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)

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arxiv 2212.05353 v2 pith:IYRXMDNV submitted 2022-12-10 math.CO

Julia Crager , Felicia Flores , Timothy E. Goldberg , Lauren L. Rose , Daniel Rose-Levine , Darrion Thornburgh , Raphael Walker This is my paper · ORCID
classification math.CO
keywords capsaffinecardcompleteevenquadstextitarbitrarycards
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abstract

We define a \textit{cap} in the affine geometry $AG(n,2)$ to be a subset in which any collection of 4 points is in general position. In this paper we classify, up to affine equivalence, all caps in $AG(n,2)$ of size $k \leq 9$. As a result, we obtain a complete characterization of caps in dimension $n \leq 6$, in particular complete and maximal caps. Since the \textit{EvenQuads} card deck is a model for $AG(6,2)$, as a consequence we determine the probability that an arbitrary $k$-card layout contains a quad.

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  1. SET! From Groups to Games

    math.HO 2025-07 conditional novelty 6.0 of 10

    A survey and extension of SET-like card games over finite groups, introducing new torsor-based games and a symmetry characterization for the group of five elements.

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