A finite-geometry instrument
Twenty Cards With Nowhere to Hide
A standard SET deck is the eighty-one points of four-dimensional arithmetic modulo three. Twenty cards can avoid every Set, but a twenty-first cannot. Hold a genuine twenty-card exception, add any remaining card, and let the browser expose the forced triple, then operate the finite-geometric counting proof that rules out every possible twenty-one-card exception.
Deck = F a Set satisfies x + y + z = zero coordinatewise
The operable base
Add the card that cannot fit
The pale tiles are an explicit cap: a layout with no Set. Choose any point absent from it and add that point. Pink marks the newcomer; every outlined triple is found from the coordinates in your browser.
The cap is ready. Choose an absent coordinate and add it.
The unique completion
Click any two outlined cards
The third is not searched for. It is recomputed as minus their coordinatewise sum.
Move the whole world
Affine maps preserve every Set
Translate, swap axes, or shear the coordinates. The tiles change, but the zero-triple count does not.
The entire outside
Not every addition fails equally
The depth layer
Completeness is not the theorem
The first bench proves that one cap cannot accept another point. It does not prove that some different twenty-one-point cap is impossible. For that global claim, suppose such a cap exists and make its hyperplanes keep three ledgers.
The sophisticated objection: perhaps the displayed cap is merely stuck. The ledger below never uses its coordinates. It applies to every hypothetical cap of the forbidden size.
Proposed hyperplane-family counts
Try to make whole counts balance
Each label records how a cap would split across three parallel hyperplanes. Counts must be whole and nonnegative.
Reader proposals refused for malformed counts:
Three ways to count the same marks
The ledgers
| Ledger | Your left side | Required | Residual |
|---|---|---|---|
| parallel families | |||
| marked pairs | |||
| marked triples |
Eliminate both end profiles
What remains
Affine classification adds context, not the proof of the bound. The cited paper reports one affine type of maximum cap in these dimensions. The first bench's maps show what affine equivalence preserves; the marked-hyperplane contradiction above establishes the upper bound without assuming that classification.
The check
The browser recomputed the construction, its complete extension sweep, the proof coefficients, and an independent integer search. These are assertions with failure paths, not a second printing of fixtures.
cards chosen from , with triples tested and violations by direct sums and by pair completion.
/ accepted by the sweep, with triples tested in each extended layout and none escaping.
integer ledgers after right-side candidates in a meet-in-the-middle search. The balanced arithmetic shadow demands families of its first profile, so the integer gate rejects it.
Davis and Maclagan, , give the exact maximum . The explicit lower bound and ledger upper bound meet there.
The checker rejects a deliberately altered tile and exposes forbidden triples. A checker that accepted it would fail this panel.
malformed cases rejected; endpoint proposals both fail rather than making the comparison vacuous.
Published dependency: a hyperplane section contains at most cap points. The paper proves that smaller-dimensional result; this browser does not exhaust all subsets of ternary three-space. Free choices: coordinate names, glyphs, tile order, default absent point, colors, and the sample affine maps. The displayed cap is generated and checked here; the classification of all maximum caps is cited, not re-proved.