kjellfp wrote:JPF wrote:In other words, is it possible for each number N such that 0<=N<=Ng, to find a (pseudo)puzzle with N solutions ?

Indeed not. The 0-clue grid has N0=6670903752021072936960 solutions, the 1-clue grid has N1=741211528002341437440. Any grid with at least two clues has less than N1 solutions, so the numbers between N0 and N1 are never the exact number of solutions to a puzzle.

Good point !

kjellfp wrote:More interresting challenge: What is the smallest number N such that no grid has exactly N solutions?

It's going to be very dificult to answer, because we will have to check every numbers from 3 to N ...

At this stage, we cannot say for instance that N =596, without making puzzles with 3, 4, ...,595 solutions.

which is a necessary, (but not a sufficient) condition.

JPF