Just for the record. I wrote, about the canonical grid,
Finally, if this grid has a 19, then all grids have a 19 (and probably an 18). This completed grid has the largest symmetry group of all grids, and there's a principle that grids with higher symmetry don't have puzzles with smaller numbers of clues. Of course I can't prove that part...
but I was wrong. Here is a grid that does not have a puzzle with 19 clues: 145726983 837495261 926381574 293874156 581269347 674153892 318547629 459632718 762918435 It does have a puzzle with 20 clues. I verified that it has no 19 with a program "checker", see http://www.maths.nuim.ie/staff/gmg/sudoku/ I've used checker to look for a 16 (which I don't believe exists). I'm currently trying to modify it to look for a pseudo-16, a puzzle with 16 clues and two completions. There is exactly one of these known so far: 5.2...4.....71...3..............46...7.2......1.......6....2.......3..1.4........ You will notice that neither 8 or 9 appears in this puzzle, so 8 and 9 can be interchanged in any completion to give another completion. That's how the two completions arise. Gary McGuire