Credit: Thomas Snyder (crazy good puzzler) was the one who discovered the two types of toroidal slitherlinks. I’ll post his solutions and mine tonight, along with an explanation of the two types of solution.
Update: it turns out this puzzle does not have a unique solution. Further, it apparently has two different kinds of solution.
I designed the solution to this puzzle to have has two distinct sides — an inside and an outside. The “outside” connects to itself around the “inside,” albeit by going out one side of the puzzle and in the other. This reminds me of an old riddle/joke: “Is a zebra black with white stripes or white with black stripes?” The answer is that the zebra is white with black stripes; notice that the belly of a zebra is solid white(more or less visible depending on the species of zebra and the picture).
Even given that, it turns out my solution isn’t unique. There is a 2 where the loop can go on either side of it. It was right there and I missed it.
But there is another way to solve the puzzle, one that has no “inside” or “outside,” or even sides at all. Imagine if the solution were a single vertical line going from the bottom of the puzzle to the top (and thereby connecting to itself). That’s a loop, but it would be possible to connect any square to any other square without crossing the loop. Hence, no sides at all. There is apparently a solution like that as well, although I haven’t seen it yet.
So I think next time I’ll have to try harder for uniqueness.
This is a different variation on slitherlinks than the last one I came up with. This is like a regular slitherlink, in that the numbers can be either inside or outside the loop. The trick here is that the puzzle wraps around: the top is connected to the bottom, and the left to the right. So while there is just one loop that doesn’t cross itself, the loop can go off one side and in the other. So this would be a valid solution:
This is a 4×4 grid. The path goes off the lower right side/in the lower left side, and then off the bottom/in the top. Note that the right and left edges are actually the same set of dots, and the top and the bottom are also just one set of dots. Hence the line at the upper right that looks stranded is actually the same line as the one to the left of the 3 in the upper left corner. Once you understand that, you can see that this puzzle has two sets of adjacent 3s: the ones side by side on the lower right, and the ones on top of each other at the lower left and upper left.
So here’s my first puzzle of this type. As usual, it has a single loop that doesn’t cross itself. Unless I’ve made a mistake there is a unique solution. Good luck!