I wrote:
The key to solving this is most probably in the Jigsaw layout.
On reflection, I realise now that this is NOT true !!
Jigsaw layouts can be powerful constraints, reducing the solution space considerably. Some Jigsaw layouts have no solutions at all.
But this particular layout is actually quite modest when measured on this scale. If we drop the NC requirement, that is, treat it as a standard Sudoku Jigsaw, and fix the contents of any one row, col, or region, then we find there are around one
billion solutions.
When we add the NC constraint, however, the effect is rather astonishing. Removing that single clue reveals that there are just
4 solutions in total:
Code:
135724968681379524246853179792418635357962481819537246463185792928641357574296813
318692475753146829297581364642735918184269753536814297971358642425973186869427531
792418635357964281813529746468375192926841357574296813139752468685137924241683579
975386142429731586864257931318692475753148629291573864647925318182469753536814297
The symmetry of the 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!
Also, if the single clue value I gave originally (4) is changed to any value other than 6, then the puzzle will have NO solution.
On reflection, then, "beyond diabolical" (for P&P solving) is probably spot on!