Puzzle 3: The One Ring

[zotmeister]

A common theme among puzzles is the creation of a single, uncrossed loop; there are at least four variations of such puzzles in popular existence. It occurred to me that there need be no reason whatsoever for these to exist as separate puzzles, and by creating the concept of the Force dot, I have merged all these very compatible puzzles together into the single one they should have been all along. The title is therefore quite fitting in multiple ways. Whether I could be said to have designed The One Ring or not is therefore debatable, but this is certainly the first of its kind.



Yet again, the left grid is an unsolved puzzle; the right grid is the unique solution. (The purple crosses are not part of the solution - only the green loop is.)

I'm going to go straight to a numbered list of rules for this one:

1) The objective is to create a linear loop of "edges", each edge connecting two orthogonally adjacent dots.
2) All the edges must be unit-length (so you can't draw an edge through spots where dots are "missing" from the otherwise evenly-spaced grid).
3) All edges must be part of the loop (so any you're given at the onset must be used).
4) There must be only one such loop.
5) The loop may not touch or cross itself at any point (which means that any dot used in the loop must have exactly two edges on it - coming and going, as it were).
6) Where numbers exist in the grid, the exact number of edges around the area the number appears in must be used as part of the loop.
7) Force dots, Angel dots, and Devil dots must be part of the loop.
8) Angel dots make order out of chaos: the loop must pass straight through Angel dots, but must bend at the next or previous dot in the loop, or both.
9) Devil dots make chaos out of order: the loop must bend at a Devil dot, but must travel straight through both the next and previous dots in the loop.

Here's a legend of my crappy pixelated dots:



So, how does one solve this? Well, it isn't tough, but it is tricky to describe, so I've buried it here if you really need it:


(Start of sample puzzle solution)
The first thing I'd do is block off the potential edges around the zero so I remember not to draw any there. I do this by "crossing off" the space the edge would go in; I have done this in purple on the sample solution. Beginners may want to also do this around the "unusuable" dot (the space in the grid) and the outer border if it helps visualization. Every time I connect two edges to a dot, I will cross off any other possible edges to that dot; that is a big help in solving. Anyway, I start in the upper-right corner. It is a Force dot, so it must be used, and as it is in the corner, it needs both edges it could have - left and down. Consider the left edge; if we were to continue down from it, the loop would then be forced right, sealing a four-edge loop in the corner! This is incorrect (we haven't satisfied all the numbers and other required dots), so we can cross off that potential edge. But now look at the dot one diagonally in from the upper-right corner - it has three directions blocked off. Since a dot used in the loop must have exactly two edges, that dot can't be used at all; we can block off the fourth direction. The loop must now continue straight left and down for another dot each. That left edge runs into an Angel dot, so it must again pass straight, but on the following dot must bend, with the only available direction being down - we cross off the leftward space and draw in the downward edge. We can also cross off down from the Angel dot and then left from there, but the next edge is not immediately obvious, so I'll just jump back to the other end of our loop-in-progress for now.

We are about to cross alongside a 1. Obviously, we can only do this once. If we go left, we'll be forced to turn down, crossing along it again, so we need to go down instead. As this satisfies the 1, we cross off the three other spaces around it. Our loop must continue down again, now parallel to the given edge. If we were to connect to that edge right now, we couldn't satisfy the 2 in the bottom-right corner - the second edge it needs would be a dead end! So instead, we continue around the 2 through the corner dot. Crossing off the two remaining spaces around the two, we can head left one more edge. Now look at the given edge: we've given each end of it only one direction to go! Each end must go left... and then we can cross off the space that would connect those two edges, as that would be a second loop. This forces the upper edge up and then left.

Now I need to be slightly clever. See that 3 in the upper-left corner? There's only one valid option for which edge it could be missing - the bottom one. If it is missing any other, there will be a dead end or two. I'll let you experiment with that yourself. Drawing in the three edges needed and crossing off the bottom space, the left edge must travel down again - right into a Devil dot. Since the loop must bend there and continue straight through the next dot, we go right twice from there - connecting into the part we drew from the given edge. The other end of the link around the 3 must now go right, connecting into the edge that bent after the Angel dot. All the edges we've now drawn are connected, and the only necessary ingredient that isn't part of the loop is the Devil dot in the bottom-left corner. Well, that's obvious to complete - two edges up and two edges right. The upper end of this must turn right, then down, then right again - if you've been vigilant in crossing off, this should be obvious - and then since we want one loop at the end rather than two, both leads travel right to complete The One Ring. ...Phew! That was a nuisance to type.
(End of sample puzzle solution)


What lies below may look like four puzzles at first glance, but it wouldn't take long to show that none of them could be individually solved. For this first The One Ring, I decided to honor the puzzles I've united by giving each of their formats their own quadrant of the grid. Rest assured, however, that it is a single puzzle. Can you find The One Ring?



As usual, the comment page is open for comments and my email inbox is open for solution verification. Of particular interest to me is what corner of this puzzle you most enjoy solving. - ZM