If a pair, a-b, is fully entwined then you can remove all digits 'a' and all but one of digit 'b' and still maintain an unique solution. If a triplet a-b-c is fully entwined, then you can remove all digits 'a' and all but one of both digits 'b' and 'c' and maintain an unique solution.

The solution to the "toughest known":

- Code: Select all
`*-----------*`

|798|635|421|

|126|974|583|

|453|218|679|

|---+---+---|

|972|586|314|

|564|123|897|

|381|497|256|

|---+---+---|

|617|352|948|

|835|749|162|

|249|861|735|

*-----------*

Out of the 36 numberpairs 20 are fully entwined (1-5, 1-6, 1-7, 1-8, 1-9, 2-3, 2-4, 2-5, 2-7, 3-4, 3-6, 3-7, 3-9, 4-6, 4-8, 5-7, 5-9, 6-8, 6-9, 8-9), two form three deadly patterns (1-2, 3-8) and one forms four deadly patterns (7-9). The rest form two.

There's also 7 triplets that are fully entwined (1-5-7, 1-6-9, 2-3-4, 2-3-7, 2-5-7, 3-6-9, 4-6-8). Removing all digits of any of these triplets would cause only 6 solutions. To compare, if you chose to remove all digits of the triplet 4-7-9, that would cause 360 solutions.

Has this field ever been researched? What's the maximum/minimum number of fully entwined pairs/triplets in a grid? What would be the average for a random grid? Can there be fully entwined quads? Does the amount of fully entwined pairs/triplets have any effect on the puzzles that can be constructed from the grid (more hard puzzles or low clue puzzles)?

I made one manual attempt to construct a grid with fewer fully entwined pairs and got 14 fully entwined pairs and 4 fully entwined triplets. I'm sure the number can be pressed down a lot from there, so I won't even bother posting the puzzle. But it seems to me the ratio would be quite high in the toughest known.

Or am I just wasting your time as there is another thread discussing this somewhere?

RW