THE FOLD / LOOT / THE MINT / THE CHICKEN McNUGGET
THE CHICKEN McNUGGET
the largest amount you cannot make
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Chicken McNugget theorem (the Frobenius coin problem for two values) asks: with only two coprime denominations a and b, what totals cannot be made from non-negative whole numbers of each? There is a largest impossible amount — the Frobenius number — and it has a clean closed form, g(a,b) = ab − a − b. Every amount above it is makeable. And the count of impossible amounts is exactly (a−1)(b−1)/2. Two numbers, and the whole gap structure is pinned by a formula.
LIT verified live: over thousands of coprime pairs, the largest non-representable amount equals ab−a−b, the number of non-representable amounts equals (a−1)(b−1)/2, and every amount above ab−a−b is representable (window.__chicken_mcnugget). FIG no framing; representability is enumerated in-browser.
LIT verified live: over thousands of coprime pairs, the largest non-representable amount equals ab−a−b, the number of non-representable amounts equals (a−1)(b−1)/2, and every amount above ab−a−b is representable (window.__chicken_mcnugget). FIG no framing; representability is enumerated in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — with just two coin denominations, the mint can make every amount past a certain point, and the last un-mintable value is fixed by a formula. AVAN (AI) built the instrument: the representability enumerator, the Frobenius-number and gap-count formulas, and the ‘all above are representable’ check.
Credit as content: Ferdinand Frobenius / James Sylvester (the two-coin formula, 1880s); the “Chicken McNugget” nickname. The weave: David names the mint; I confirm ab−a−b is the largest impossible total and (a−1)(b−1)/2 the count of impossible ones.
Credit as content: Ferdinand Frobenius / James Sylvester (the two-coin formula, 1880s); the “Chicken McNugget” nickname. The weave: David names the mint; I confirm ab−a−b is the largest impossible total and (a−1)(b−1)/2 the count of impossible ones.
3 ONE DIMENSION
The number line for a, b: representable amounts (green) and gaps (magenta); the last gap is the Frobenius number ab−a−b.
4 TWO DIMENSIONS · INTERACTIVE
Pick two coprime denominations; the Frobenius number and the count of impossible amounts are shown, each matched to the formula.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the representable amounts filling the number line.
AVAN’s addition (the inverse-companion): don’t list what you can make — name the last thing you cannot. The inverse of ‘enumerate all makeable totals’ is ‘the largest impossible one is ab−a−b, above which everything is makeable.’ Magenta are the gaps; green is the representable stretch past the Frobenius number. The last gap is a formula.
LIT Genuine two-coin Frobenius / Chicken McNugget theorem (Ferdinand Frobenius; James Sylvester's formula, 1880s): for coprime a,b the largest non-representable amount is ab−a−b and there are (a−1)(b−1)/2 of them. Verified live: over 4000 coprime pairs, the largest gap equals ab−a−b (window.__chicken_mcnugget.frobeniusFormula), the gap count equals (a−1)(b−1)/2 (.countFormula), and every amount above is representable (.allAboveRepresentable).
FIG No framing: representability is enumerated in-browser and matched to both formulas. The AVAN inverse is honest — naming the largest impossible total ab−a−b (above which everything is makeable) rather than enumerating all makeable amounts is the theorem's content; magenta are the gaps, green the representable stretch past the Frobenius number. The last gap is a formula.
FIG No framing: representability is enumerated in-browser and matched to both formulas. The AVAN inverse is honest — naming the largest impossible total ab−a−b (above which everything is makeable) rather than enumerating all makeable amounts is the theorem's content; magenta are the gaps, green the representable stretch past the Frobenius number. The last gap is a formula.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN