How to Solve Hard Loop Puzzles (Slitherlink)

3 ways to solve

These are listed from the easiest to spot. The number on the right counts the free puzzles, worked through in this page's order, that stopped at the ways before it and then had their next line or cross decided by this one. Inside and outside is counted on Hard, and trying a line on Oni.

1A line through a corner

A dot has 0 or 2 lines. Take the 2 squares diagonally opposite each other across a dot, and pair up the 2 sides of each that touch the dot. If one pair has exactly 1 line, the other pair has exactly 1 too, because the loop passes through the dot to the far side. If the dot is on the edge of the board and the far side has only 1 side, that side is a line.

Used at a number's corner, this joins lines diagonally. When exactly 1 line comes into a corner of a 1 or a 3, it is the same as the patterns "A line into a 1's corner" and "A line into a 3's corner": the 2 sides at the opposite corner are crossed for a 1 and lines for a 3. What is new is the 2. If a corner of a 2 has exactly 1 line, the other 2 sides have only 1 as well, so the opposite corner has exactly 1 too. The line passes diagonally through the 2 and on to the dot beyond.

1. The 1 at row 8, column 1: its top side is a line and its right side is crossed. The top line has come as far as the 1's top right dot. A dot has 2 lines, so it goes on to 1 of the dot's remaining 2 sides (the bottom or left of ②)
2. The 2 at row 7, column 2: its bottom left corner has exactly 1 line, so its top right corner (top and right) has exactly 1 too. That line passes through the top right dot
3. The 3 at row 6, column 3: exactly 1 line comes into its bottom left corner (bottom and left). The 3 can use only 1 side there, so its top and right sides are lines

Play Hard No.5Print (with answer)

A dot takes at most 2 lines

A relative of the line through a corner, this works even when you cannot tell it is "exactly 1". A dot takes at most 2 lines, so if a corner has 1 or more lines, the corner diagonally opposite across the dot can take at most 1. A corner of a 3 always has at least 1 line, so beyond a corner of a 3 it is always "at most 1". A 2 with at most 1 at one corner uses 1 or more at the opposite corner. With 1 or more at one corner, the opposite corner has at most 1.

The pattern "3 and 3 on a diagonal with 2s between" comes from this idea too.

1. Row 8, column 6: the line on the top side has reached the dot at the right end. A dot takes at most 2 lines, so the 2 sides beyond the dot (the bottom and left of ②) can take at most 1
2. The 2 at row 7, column 7: it uses 2 lines, so the top and right sides left must take 1 or more. The top side is crossed, so the right side is a line

Play Normal No.10Print (with answer)

The ways so far (counting, patterns, a line through a corner, a dot takes at most 2 lines) solve all 30 free Normal puzzles to the end.

2Inside and outside

Spread from the edge of the board

Even partway through, you can tell inside from outside for some squares. First look at the edge of the board. A square is inside if its edge side is a line and outside if its edge side is crossed. From there, spread to the neighbouring squares. The neighbour across a cross is on the same side, and the neighbour across a line is on the opposite side.

Counting, patterns and lines through corners are all used up here. At the top right of the board, spread inside and outside.

1. Row 3, column 10: its edge side (right) is a line, so inside
2. Row 3, column 9: across a cross from ①, so inside, the same as ①
3. Row 2, column 9: across a line from ②, so the opposite: outside
4. Row 2, column 8: across a line from ③, so the opposite: inside
5. Row 1, column 8: across a cross from ④, so the same: inside

⑤ is inside, but above it is beyond the board (always outside). Inside meets outside there, so the top side of ⑤ is a line.

Play Hard No.16Print (with answer)

A line where inside meets outside

As you spread from the edge of the board, an inside and an outside that came from different places can end up side by side. The side between them is a line.

1. Row 7, column 1: its edge side (left) is a line, so inside
2. Row 7, column 2: across a cross from ①, so the same: inside
3. Row 10, column 2: its edge side (bottom) is crossed, so outside
4. Row 9, column 2: across a line from ③, so the opposite: inside
5. Row 8, column 2: across a line from ④, so the opposite: outside

② inside and ⑤ outside are neighbours, so the side between them is a line.

Play Hard No.6Print (with answer)

Numbers and inside and outside

This also works on a number you cannot yet place as inside or outside. If the squares left and right of a 1 are both outside, then a 1 that was inside would have lines on those 2 sides, more than 1. So the 1 is outside and its left and right sides are crossed. If the squares left and right of a 3 are both outside, then the 3 is the opposite, inside, and its left and right sides are lines. It need not be left and right: any 2 squares known to be on the same side decide it the same way. Between inside squares, a 1 is inside and a 3 is outside.

1. Row 10, column 7: its edge side (bottom) is a line, so inside
2. The 3 at row 5, column 7: 5 crosses run upward from ①, so the same: inside
3. Row 5, column 8: across a line from ②, so the opposite: outside
4. Row 5, column 10: its edge side (right) is crossed, so outside
5. The 1 at row 5, column 9: none of its 4 sides is decided, but ③ and ④ on its left and right are both outside. If the 1 were inside, those 2 sides would be lines, more than 1, so the 1 is outside. Its left and right sides are crossed

Play Hard No.17Print (with answer)

3Trying a line

On an Oni puzzle there are moments when none of the ways so far takes you further. Then draw a line on a side for a moment and follow what it decides. If something cannot be - a number with too many lines, 3 lines at a dot, a small loop closing - that side is crossed. You can try a cross the same way.

1. Try a line on the top side of row 6, column 4
2. Left end dot: the other 2 sides (the bottom and right of the 0 at row 5, column 3) are crossed, so the line turns down and runs along the left side of row 6, column 4
3. Right end dot: the bottom of the 1 at row 5, column 5 and the right of row 5, column 4 are crossed, so the line goes down the right side of row 6, column 4
4. Bottom left dot: the bottom of row 6, column 3 and the right of row 7, column 3 are crossed, so the line turns right and runs along the bottom side of row 6, column 4

That closes a small loop around row 6, column 4. These 4 sides are decided together, so trying a line shows it at a glance here. On other puzzles you may have to follow it much further.

A small loop cannot form, so the top side of row 6, column 4 is crossed. From here you can go on again with counting and patterns.

Play Oni No.5Print (with answer)

Where to try

• Pick a side near the end of a line, or where few sides are undecided. Following it a little soon shows something that cannot be.
• Try 1 place at a time. If a line fails, the side is crossed; if a cross fails, the side is a line.
• If nothing happens after several moves, take the tried line off and try another side. In the game, Undo takes it off.

Questions