After a break over the weekend I tried the puzzle again, and solved it in a couple of minutes. I chose to start with the 6 at the bottom-left and try connecting them directly, since they were close. Then I was forced to extend the 1, so chose the quickest path for that too, wrapping off the bottom beneath the 7. From there I pushed the 7 down and round via 3 corners to meet up, and then the 5 is forced. From there there are only a couple of choices, one of which finishes the puzzle successfully.

The edges are obviously a great help in a normal puzzle, and you can massively prune the search space by considering non-uniqueness as you extend lines. For example in this puzzle I found almost any moves I made soon proved that I had clearly made any potential solution non-unique, since of course lines cannot ever ‘touch themselves’ in an adjacent square and still be part of a unique solution. Therefore it’s also the case that any empty space between lines/clues which would require a line to bend back on itself or ‘touch itself’ to fill it must also be part of a non-unique solution, and so the lines that created that space must be wrong too.

In regular puzzles you can also start by considering numbers with both pairs close to the edges, since these may well simply go around the edge in some way. Looking for likely crossing/non-crossing pairs also helps.

But most of that breaks down in this toroidal puzzle. Essentially I think it requires a trial+error approach which can be followed reasonably quickly only by applying a keen eye to what must cause non-uniqueness!

Tried figuring it out but felt as if I was guessing most of the time. So my question is, what are some logic rules to get going? With the other ones the strategy is to start from the corners and sides and work inwards. So without the walls, where is the logic to start (the 4s or 2s)?

The really good thing about this hard puzzle is it caused me to forget about it and complete some tasks this weekend that really needed to get done.

I’d probably like these puzzles more if I had more methods of how to do them.