Sylver Coinage — play the exact solver

New here? Watch the story, then play the lessons

The Sylver Coinage Story is a six-minute animated history: Sylvester's coins (1884), Winning Ways (1982), George Sicherman's papers and tables, Thomas Blok's reports, and the 2026 machine-checked campaign. The Sylver Mint retraces it in eleven small games, from playing a perfect opponent to assaying a Quiet End certificate.

Play against perfect play

Start from a position (comma-separated generators, gcd 1 — try 4,5, 5,7, 6,7, 7,8,13, or the deep 16,20,28,35,54):

Start game Just evaluate

The browser solver handles Frobenius numbers up to ~140 comfortably (it is the Python reference's algorithm, ported and differentially tested to matching state counts). Positions with gcd > 1 have infinitely many legal odd moves and are settled by the campaign's theorems rather than finite search — query those against the cache below.

Query the campaign cache

The campaign proved 273,000+ exact outcomes (over 10¹² evaluated states), many far beyond browser reach. This fetches the audited cache straight from the repository and looks positions up by their minimal generating set.

Load campaign cache (~10 MB)
Look up

The certified opening-16 table

After the opening move 16, every even reply from 2 through 24 has a certified winning answer (odd replies lose by Hutchings' theorem):

Beyond the table, the campaign records answer 26 by 88 (U), 28 by 58 (Y) and 30 by 56 (Z), the node T answers 34 by 20, and the finite P-position {16,23,36} answers 36 by 23; 32 is not a legal reply. The lowest reply to 16 still without a certified answer is 38 (the answer ledger). By search, 72 answers it, so the lowest reply with no known answer is 40 (the search for an answer to 38). See the research log — and browse the auto-generated campaign tables in the style of Blok's 6-16 Tables.

The Book

The cheapest-to-check certificate found for each certified result of the campaign, from the certificate-golf arena. Open the Book and replay any small certificate in your browser with an independent JavaScript checker.

W = {16,26,62,98} is P

Every one of W's 52 Quiet End obligations is answered, so W is a P-position and Q = {16,26,88,98} is N. See each obligation's evidence and replay the smaller Book certificates among them in your browser.

U = {16,26,88} is P: the reply 26 to 16 loses

After 16 and 26, naming 88 reaches U. The last of U's 59 Quiet End obligations, the move to X = {16,26,82,88}, is covered: X's reply 701 reaches a P-position, confirmed by two sequential runs of engines with independent move code that agree on the exact state count (in place of native and Python replays, which would not fit the host). So U is P and 88 answers 26. See U's obligations.

Y = {16,28,58} is P: the reply 28 to 16 loses

After 16 and 28, naming 58 reaches Y. All 54 of Y's Quiet End obligations are covered: 48 by finite witnesses replayed natively and in Python, six by certified P-positions. So every reply to 16 up to 28 now has an answer. See Y's obligations.

Z = {16,30,56} is P: the reply 30 to 16 loses

After 16 and 30, naming 56 reaches Z. All 52 of Z's obligations are covered, one through a second P-position, Z′ = {16,30,40,44}, which this confirms independently (it is on Sicherman's list). Every reply to 16 up to 30 now has an answer. See Z's obligations.

Reply 38: four P-positions refute five answers

38 is the lowest reply to 16 without a certified answer. Four short P-positions, I = {16,20,22,24}, B28 = {16,28,38,40}, B24 = {16,24,38,44} and B56 = {16,38,56,60}, show that 24, 28, 40, 44 and 56 do not answer it. I and B24 were already on Sicherman's list, and are confirmed here; B28 and B56 are not on it. See their obligations.

Reply 38: by search, 72 answers it

On October 18 a week-long search finished: by search, {16,38,72} is P, since each of its 68 obligations is N; the last to fall was {16,38,72,78}, which 367 wins. So, by search, the reply 38 to 16 loses, and the lowest reply with no known answer is 40. It is not yet a certificate: 61 of the 68 obligations are now certified, and the other seven rest on search. See the record.