We still haven't proven whether it's possible to construct a complete and general theory predicting the exact number of distinct solutions to any well-defined combinatorial puzzle.
open
Global / Unspecified, Global
WS01355
Many specific combinatorial puzzles have been solved exactly, but a fully general theory predicting solution counts across every possible puzzle type remains an unresolved and active area of combinatorics. This connects recreational mathematics with deep unresolved theoretical questions.