Fifth in a series with puzzle solving tips. This time, with some advice on global constraints that arise in loop puzzles.

Some of you have asked why I don’t put “general tips” up before I post any puzzles. I find that learning how to solve puzzles is more interesting than reading how to do every single step. I won’t post a list of the top 20 Slitherlink patterns you should memorize, for example, on the rules and info page. Most of you would probably get more enjoyment figuring those out for yourself. But, after you’ve solved a challenge, I do like highlighting some of the things as constructor I tried to include, perhaps as a lesson for the future. This is what I will do this week with parity constraints for the Friday Masyu.

Basic Masyu steps, particularly regarding black circles and edges, will get you to this state. There is one more extra line drawn, from the black circle highlighted in yellow. Notice that if this connection went up instead of down it would immediately close a loop. This is the most basic loop closure step to observe in this kind of puzzle and should become close to automatic with practice.

Now we get to see loop parity at work. The general idea is that until you’ve made all the connections, in any isolated region there must always be an even number of loose ends. When it looks like you will strand an odd number of ends, you have a problem. And when you have to choose between ends to enter into a region, you must always choose the end that will still let you form a single loop and not two or more loops.

Consider the lower-right region highlighted in yellow in the next figure. There is one end, A, already in there. There are three other ends, B, C, and D, that can reach into the region. To have a single loop, either 2 or 4 of these ends must connect into the yellow space. This is the basic parity constraint.

Notice that both C and D cannot get to yellow, as if D goes to the right it blocks C, and if C goes down it blocks D. So there are only 2 and not 4 ends that reach into yellow. Now you have to choose against the end that will give you more than one loop. Since C is connected to A, it cannot be the extra end in the yellow region. Draw a vertical connection from C up (as shown in red). This deduction will lead to a lot more progress.

Basic Masyu steps get you to this next stall point. Again, parity is important but in a slightly different way. Here you need to create a new strand to restore parity. Notice how in the yellow region there are just 3 ends. There must be another end that gets into this box. Not knowing how it connects yet, you can still deduce there is a line in red going up to the top-left corner. Continuing that connection to the top-middle of the grid will lead to the final connections needed to complete the puzzle.

While there are far harder observations about parity that come up in loop puzzles, even these simple ones can slow a majority of solvers who aren’t very familiar with them. Sure, bifurcation/guessing could have gotten you through this 10×10 puzzle, but on a larger puzzle you won’t want to go so far down a guessing path when “simple” deductions about parity can give you significant progress.