Thinking ahead without guessing

Sooner or later every path end on the board has two or more open sides, and no cell is squeezed. Many players start guessing here. Instead, pick a path end with exactly two choices, pretend you've made one of them, and follow only the moves that choice forces. If the board breaks, the choice was wrong and the other one is proven.

The puzzle

Three forced moves come first. Orange's top dot at row 1, column 6 can only go down. Red's dot at row 5, column 1 can only go up, because Purple's dots sit to its right and below. Purple's corner dot then has one open side, so Purple steps right to row 6, column 2, next to its other dot.

Nothing else is forced now. Look at Orange's lower dot at row 3, column 2. It has Green dots above and to its right, so it has two choices, left to row 3, column 1, or down to row 4, column 2.

Following one option

Try left. From here on, only make moves that are forced.

1. At row 3, column 1, Orange's only open side is up, because Red's tip is below. Orange goes up to row 2, column 1.
2. From there it must go up again into the corner, then right along the top edge to row 1, column 3. At row 1, column 2 the cell below is a Green dot, so there's no turning down.
3. Orange now surrounds Green's dot at row 2, column 2 on three sides. That dot's only open side is to its right, so Green steps to row 2, column 3, next to its other dot.
4. That takes away Orange's way down at row 1, column 3. Orange goes right to row 1, column 4, and then down to row 2, column 4, because row 1, column 5 is Blue's dot.
5. Blue's dot at row 1, column 5 is now hemmed in on the left by Orange. Its only open side is down, so Blue goes to row 2, column 5, and from there down again to row 3, column 5.

Now look at Orange's two tips at row 2, column 4 and row 2, column 6. Blue's path runs down between them. The tip at row 2, column 4 has one open cell below it, and that cell is boxed in by a Green dot, a Red dot and Blue's tip. Orange has no route between its ends. In the language of the bottlenecks lesson, Orange is cut off.

The site's puzzle walkthroughs are drafted from a deduction log written by its logic solver. For this puzzle, the log records the step like this: "Orange has more than one option from row 3, column 2. Going left fails because Orange would be cut off from its partner. So Orange goes down one, right one, down two and onward."

Orange had only two options, so the failure of one proves the other. Orange goes down, and the rest of the puzzle follows from forced moves.

The cells Orange would have used in the trial end up belonging to Red, which loops around the top-left to reach its dot at row 4, column 4.

Why this isn't guessing

A guess commits to a move without knowing whether it's right, and finds out later. Thinking ahead commits to nothing until a move is proven wrong. Every step inside the trial was forced by the step before it, so the contradiction at the end traces straight back to the first move. That is a proof by contradiction. If the first move were part of the solution, every forced reply would be too, and the solution can't contain a cut-off color.

Two rules keep it honest:

• Only follow forced moves inside a trial. If you make a free choice along the way and then hit a contradiction, you only know that the combination fails. The first move might still be right.
• A trial that survives proves nothing. If you follow the forced moves and the board stays healthy without being finished, you have learned nothing about whether the first move is right. Don't keep it. Undo it and look elsewhere.

How far to look

The failures to watch for are the ones from earlier lessons, a dead end, a trapped cell, a sealed pocket, a cut-off color or a boxed-in tip. Check for them after every forced reply, as well as at the end.

The example above took nine forced moves, but they came in long runs along one path, which makes them easy to follow. A good stopping point is when the forced moves run out. If nothing has broken by then, the trial has told you all it can at this depth.

The site's walkthroughs sometimes go one level further. When a trial runs out of forced moves without breaking, the solver looks inside the trial for a path end whose options all fail when tested the same way. If it finds one, the original move is wrong. The walkthrough text for that reads "a few forced moves later, Blue would have no option that works". Puzzles that need this second level are rated Advanced or Expert on this site. It's hard to hold in your head, so try every one-level trial on the board first.

Keeping track

The difficulty is remembering which cells are filled only in your imagination. Some habits help:

• Follow one path at a time. Forced moves often come in runs along the same color, as Orange's did here. Finish a run before checking other colors.
• Recount after each step. The cells that matter are next to the newest tip and the cell it just left.
• Restart instead of patching. If you lose track, start the trial again from the beginning. Patching a half-remembered position is how a trial turns into a guess.

If you play in an app that lets you undo, you can draw the trial for real and undo it afterwards. The logic is the same, as long as you only draw forced moves.

When to look at the other option instead

Which option you test first matters, because a failing trial is quicker to finish than a surviving one. Choose the option most likely to cause damage:

• the one that moves toward a crowded area full of dots,
• the one that hugs an edge or a finished path and could seal something off,
• the one that crosses a narrow gap another color needs.

In the example, going left headed into a corner next to two Green dots and a Red tip, which made it the obvious one to test. Going down led into open space, where a trial tends to run on without breaking.

If your first trial survives, test the other option. If that one fails, you're done. If both survive, neither is proven, so move on to a different path end or a different part of the board. Some positions need a different idea entirely, such as the uniqueness trick. For practice, the thinking ahead practice set collects puzzles that need these trials.