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

TL;DR

This paper classifies caps in affine geometry AG(n,2) up to size 9, characterizes caps in dimensions up to 6, and applies these results to analyze card layouts in the game EvenQuads, determining probabilities of containing quads.

Contribution

It provides a complete classification of small caps in AG(n,2) and applies this to analyze card layouts in EvenQuads, linking geometric structures to game probabilities.

Findings

Classified all caps of size ≤ 9 in AG(n,2).

Characterized caps in dimensions up to 6, including maximal caps.

Determined probability that a random k-card layout contains a quad.

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.