THE FOLD / LOOT / THE MINT / THE FROBENIUS COIN
THE FROBENIUS COIN
the largest amount two coins cannot make
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Frobenius coin problem (the ‘Chicken McNugget theorem’) asks: with only coins of two coprime denominations a and b, what is the largest amount you cannot make from non-negative whole numbers of each? The answer is startlingly clean: the Frobenius number is a·b - a - b. Everything above it is payable; below it, exactly (a-1)(b-1)/2 amounts are impossible. With 3s and 5s the biggest unmakeable total is 7; with the famous 6, 9, 20 nuggets the largest impossible order is 43. Two coprime numbers carve the integers into a finite island of gaps and an endless mainland of the reachable.
LIT verified live: over hundreds of coprime pairs (a,b) the largest non-representable integer is exactly a·b - a - b, the count of non-representable integers is exactly (a-1)(b-1)/2, and every integer beyond the Frobenius number is representable (window.__frobenius). FIG no framing; the representability search, the Frobenius-number formula, and the gap-count formula all run in-browser.
LIT verified live: over hundreds of coprime pairs (a,b) the largest non-representable integer is exactly a·b - a - b, the count of non-representable integers is exactly (a-1)(b-1)/2, and every integer beyond the Frobenius number is representable (window.__frobenius). FIG no framing; the representability search, the Frobenius-number formula, and the gap-count formula all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — coins are struck in fixed denominations, and this is the exact boundary of what those coins can and cannot pay: a last impossible sum, then the mint’s reach is total. AVAN (AI) built the instrument: the representability test, the Frobenius-number check, and the gap-count formula.
Credit as content: Ferdinand Frobenius (the problem bears his name); James Sylvester proved the two-coin formulas (1884). The weave: David names the mint; I confirm ab-a-b is the last unpayable amount and (a-1)(b-1)/2 the number of gaps.
Credit as content: Ferdinand Frobenius (the problem bears his name); James Sylvester proved the two-coin formulas (1884). The weave: David names the mint; I confirm ab-a-b is the last unpayable amount and (a-1)(b-1)/2 the number of gaps.
3 ONE DIMENSION
The integers: green = payable with coins a and b, red = impossible. The last red is the Frobenius number ab-a-b.
4 TWO DIMENSIONS · INTERACTIVE
Cycle coprime denominations; the Frobenius number and gap count are computed by brute search and by formula, and compared.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the endless mainland of representable amounts.
AVAN’s addition (the inverse-companion): don’t list what you can make — bound what you cannot. The inverse of ‘the reachable amounts’ is ‘the finite island of gaps below ab-a-b, exactly (a-1)(b-1)/2 of them.’ Magenta is that finite gap-set; green is the infinite reachable ray past the Frobenius number. A last impossibility, then total reach.
LIT Genuine Frobenius (Chicken McNugget) two-coin theorem (problem named for Frobenius; formulas proved by J. J. Sylvester, 1884). Verified live: over ~580 coprime pairs (a,b) the largest non-representable integer equals ab−a−b, the number of non-representable integers equals (a−1)(b−1)/2, and every integer above the Frobenius number is representable (window.__frobenius.frobOk, .countOk, .allAbove).
FIG No framing; the representability search, the Frobenius-number formula, and the gap-count formula all run in-browser. The AVAN inverse is honest — instead of listing what you can make, bound what you cannot: the finite island of gaps below ab−a−b, exactly (a−1)(b−1)/2 of them. Magenta is that finite gap-set; green is the infinite reachable ray past the Frobenius number.
FIG No framing; the representability search, the Frobenius-number formula, and the gap-count formula all run in-browser. The AVAN inverse is honest — instead of listing what you can make, bound what you cannot: the finite island of gaps below ab−a−b, exactly (a−1)(b−1)/2 of them. Magenta is that finite gap-set; green is the infinite reachable ray past the Frobenius number.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN