Jigsaw Layouts (JL's) can be powerful constraints, reducing the solution space considerably compared to standard Sudoku with its 3x3 boxes. There is a distinct "catalog" of solutions for each different JL. Some Jigsaw layouts have no solutions at all, so for these NED = NSG = 0. One pet project of mine is to find some method of predicting which JL's have solutions, and which JL's are more constrained, etc. (By means other than solving them!)
Non-consecutive (NC) is another powerful constraint. For standard Sudoku, adding the NC constraint reduces the NED count by a factor of roughly 16,500, to just 330,845. Variant NC+ ("cyclic NC") adds the extra forbidden pair {1, 9}, and here NED is reduced to just 12,263.
The NC/NC+ variants also have the interesting property that the minimum number of clues required for a valid puzzle is greatly reduced, well below the 8-clue theoretical limit that applies to "normal" variants (ie SudokuP, SudokuX, SudokuJ etc without cell value restrictions). For standard Sudoku + NC, tarek and I have recently found 2 and 3 clue puzzles.
The recently discovered 1-clue puzzle is a "SudokuJ-NC" (a standard Jigsaw + NC constraints).
The Jigsaw layout for the 1-clue puzzle is actually not particularly constrained - solving it as a standard Sudoku Jigsaw, with one row, col, or region fixed, still gives over 1 billion solutions. But it is just enough for NC to give the 1-clue puzzle.
I have since learned that this puzzle, with no clues at all, has NSG = 4. The complete catalog is:
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135724968681379524246853179792418635357962481819537246463185792928641357574296813
318692475753146829297581364642735918184269753536814297971358642425973186869427531
792418635357964281813529746468375192926841357574296813139752468685137924241683579
975386142429731586864257931318692475753148629291573864647925318182469753536814297
The symmetry of the Jigsaw layout means that solutions #1 and #2 are essentially the same, just mirror images. The same goes for #3 and #4.
The only relabelling allowed under NC rules is a reversal of the cell values, ie {123456789} is replaced by {987654321}, so solution #1 and solution #4 are essentially the same, and so are #2 and #3.
So in fact there is only one "essentially different" solution: NED = 1.
) This is something that many have been trying to do for ages!!!
