Sylver Coinage name a coin nobody can make change for
John Conway's game. Take turns naming a positive whole number that cannot be made as a sum of numbers already named (repeats allowed). Whoever names 1 loses. Once the named numbers have gcd 1 only finitely many numbers are left, and this page solves those positions exactly: legal moves are marked ✓ (wins against any defence) or ✗. Before that, infinitely many numbers are legal and only theory can help — and theory has a hole: nobody knows whether the opening 16 wins.
position
who wins from here
moves
Why the game always ends, and when it becomes finite
While the named numbers share a factor g, every number not divisible by g is still legal — infinitely many moves. Only finitely many multiples of g can be named before g must drop, so every game ends. Once g = 1 the numbers still legal are the gaps of a numerical semigroup; the largest one is its Frobenius number t (for two coprime numbers a, b it is Sylvester's (a−1)(b−1)−1). The solver stores a position as the least member of each residue class mod a fixed named number (its Apéry set), so a position with t in the hundreds is a dozen small integers.
Enders and a strategy nobody can write down
A position is an ender when every legal move except t rules out t. Then the player to move must win: if some other move wins, play it; if not, play t — every reply is then a move that would have been legal (and losing) anyway. Hutchings showed every coprime pair {m, n} is an ender, so after a prime opening p ≥ 5 every reply hands the opener a won ender: p wins, but the proof does not say how. Sicherman's quiet enders are the ones where t − n is ruled out for every legal n — exactly those with t odd.