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<p>The card game <i>Quads</i> provides a concrete model for the affine geometry <i>AG</i>(<i>n</i>,&#xa0;2), where a <i>quad</i> corresponds to four points that sum to zero. Motivated by this connection, we study quad-free subsets of <i>AG</i>(7,&#xa0;2), called <i>caps</i>. We provide an explicit classification, up to affine equivalence, of all caps of size <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(k \ge 10\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation>, showing that there are two equivalence classes of 10-caps and one each for 11-caps and 12-caps, with 12 being the maximum possible size. This classification is obtained by constructing explicit, affine-invariant structural templates arising from distinct basis types within each equivalence class, which are then used to enumerate caps of a given size. As an application, we compute the probability that a random <i>k</i>-card layout contains a quad.</p>

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How Many Cards Should You Lay Out in Quad-128: A Classification of Caps in \({{\,\textrm{AG}\,}}(7,2)\)

  • 期刊论文

  • 发表日期：

    2026年4月13日

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  • Kariane Calta,
  • Timothy E. Goldberg,
  • Dyana R. Harrelson,
  • Lauren L. Rose

摘要

The card game Quads provides a concrete model for the affine geometry AG(n, 2), where a quad corresponds to four points that sum to zero. Motivated by this connection, we study quad-free subsets of AG(7, 2), called caps. We provide an explicit classification, up to affine equivalence, of all caps of size \(k \ge 10\) k ≥ 10 , showing that there are two equivalence classes of 10-caps and one each for 11-caps and 12-caps, with 12 being the maximum possible size. This classification is obtained by constructing explicit, affine-invariant structural templates arising from distinct basis types within each equivalence class, which are then used to enumerate caps of a given size. As an application, we compute the probability that a random k-card layout contains a quad.