Slitherlink

Rules

1.Draw one single closed loop along the dotted grid lines, from dot to dot, horizontally or vertically.

2.The loop never crosses itself and never branches. At every dot, either two of its lines meet, or none.

3.A number in a cell says exactly how many of the four sides of that cell belong to the loop: 0, 1, 2 or 3.

4.A cell without a number sets no condition. The loop may use any number of its sides, from none to four. An empty cell does not mean the loop has to stay away.

5.When you are done, all your lines form one connected ring, not two or more separate ones. There is exactly one loop that satisfies all of this.

How to solve

Draw lines, and mark every side that cannot belong to the loop with a small x. This is the whole trick. The crossing rule below works by counting lines and x's at a dot; without the x's on paper there is nothing to count, and the puzzle looks stuck when it is not.

1.Start at the 0s. A 0 puts an x on all four of its sides. Every puzzle in this book opens there.

2.Count at the numbers. If a number already has its lines, x its remaining sides. If it has exactly as many open sides as it still needs, draw them all. A 3 next to an x gets its other three sides at once.

3.Count at the crossings. Look at a dot and the sides that meet there. Three cases decide something: two lines already meet, so x the rest; one line ends there and only one side is open, so the line continues along it; no line arrives and only one side is open, so that side is an x, because a single line would be a dead end.

The three cases at a crossing, from left to right, each before and after:

Most of the work happens at the crossings, not at the numbers. Measured over ten Easy puzzles of this book: 386 of 601 deductions, about two in three, came from rule 3.

4.Corners and diagonal 3s. At a dot beside a number, the loop can only pass in a few ways, and whatever all of them agree on is certain. The patterns you will meet most: a 1 in a corner of the grid puts an x on both corner sides; a 3 in a corner of the grid gets both corner sides drawn; two 3s touching diagonally each get the two sides that face away from the shared corner.

5.No early loop. A side that would close a ring while other lines are still outside it is an x. Example: the two ends of one line piece sit at dots next to each other, and there is a line somewhere else as well. The side between those two ends would close the piece into a ring, so it gets an x.

6.Follow it through. Draw one open side in your head, apply 1 to 5, and if that leads to a contradiction, the side is an x. Never more than one step deep.

The level tells you what you need: Easy uses 1 to 4, Medium adds 5, Hard adds 6. You never have to guess and undo.

Worked example

A 6×6 grid, Easy, 15 numbers. Solve along with the text. The small numbers at the edge name the rows and columns. Lines are black, an x marks a side that is ruled out, and a gray shade shows what is new in each picture.

Steps 1 to 2, rule 1. The 0s first: row 1, column 6; row 3, column 6. Each gets an x on every side that does not have one yet. These are the first x's, and they cost nothing.

Step 3, rule 3. No line reaches the top left corner of row 1, column 6 and only one side is still open there. A single line would be a dead end: put an x on it.

Step 4, rule 2. The 2 in row 1, column 5 still needs 2, and exactly two sides are open. Draw them all.

Step 5, rule 3. Two lines already meet at the top left corner of row 2, column 5. Put an x on the rest.

Step 6, rule 3. A line ends at the top left corner of row 1, column 5 and only one side is still open there. The line has to continue that way.

Steps 7 to 8: rules already shown, 2 times.

Step 9, rule 2. The 1 in row 3, column 5 already has its line. Put an x on the remaining two sides.

Steps 10 to 13: rules already shown, 4 times.

Step 14, rule 4. No simple rule applies here. At the top left corner of row 5, column 6 the loop can pass in only a few ways that fit the numbers around it. All of them agree on three sides. The crossing is circled, and the sides all ways agree on are shaded.

The remaining 42 steps (15 to 56) need nothing new. Of all 56 steps in this example, 39 come from the crossing rule. The finished loop: