Short answer: Nobody knows (as of 2025).
Details:
- The opening move 16 in Sylver Coinage is a famous open problem. It has been studied since Conway introduced the game, and it is still not known whether the first player can force a win after opening with 16, assuming optimal play by both players.
- It is widely believed (but unproven) that 16 is a first-player win.
- A standard reduction of the question is this: after 16, the only plausible reply for the second player that is not known to lose is 5. All other replies are known to allow the first player to force a win. Thus the unresolved core is the position with moves {16,5} on the table and the first player to move.
- Many partial structural results are known (e.g., openings of the form 5n are first-player wins; much is known for openings congruent to 2 or 3 mod 5; small cases like 6 and 11 are settled), but 16 (the smallest 5n+1 beyond those) has resisted complete analysis.
References:
- E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways for your Mathematical Plays, 2nd ed., Vol. 1, chapter on Sylver Coinage. They single out 16 as the least unresolved opening and discuss why 5 is the only serious reply.
- R. K. Guy, Unsolved Problems in Combinatorial Games (in Games of No Chance, MSRI Publications). The “16 problem” is explicitly listed as an open problem.
On complexity:
- For a fixed opening number like 16, “PSPACE-hardness” is not a meaningful notion (it’s a single instance). For the general decision problem “given a finite set of previously named integers, decide whether the next player has a win,” there appears to be no established complexity classification in the literature; in particular, PSPACE-hardness for Sylver Coinage is not known.
