Easy
Zero Clue
A 0 means no edge of this cell can be part of the loop — mark all four sides with an X.
Full Clue
A 4 means every edge of this cell is part of the loop — draw all four sides as lines.
Clue Complete
Once a clue already has all its required lines, every undecided side around it is an X.
Clue Needs Every Side
When a clue still needs exactly as many lines as it has undecided sides, every one of those sides is a line.
Low Corner Clue
A 0 or 1 in a grid corner rules out both outer sides, because those two sides must match.
High Corner Clue
A 3 in a grid corner forces both outer sides to be lines, because the two inner sides alone cannot reach 3.
Dot Complete
Once two lines meet at a dot, every other edge touching that dot is an X.
Continue the Line
One line reaches a dot and only one exit remains, so the line must continue through that exit.
No Single Edge
An untouched dot with only one possible edge cannot use it, because that would leave exactly one line at the dot.
Medium
Premature Loop
If an edge would close a smaller loop while lines or clues remain outside it, that edge is an X.
Final Closure
When closing the path here satisfies every clue and accounts for the whole loop, the closing edge is a line.
Border 1-3
A 1 and 3 sharing a border dot force edges that the same pair in the board's interior would leave open.
Border Clue and Dot
One clue combines with a nearby border dot, whose missing outside edge reduces its possibilities.
Adjacent 3s
Two 3s side by side: the outer edge of each one — the side facing away from its partner — is always a line.
0 Beside a 3
A 0 next to a 3 settles the 3 outright: the shared edge is blank, so the 3's other three sides are all lines.
Hard
Diagonal 3-1 Transfer
A corner 3 transfers one required line through a shared dot, using up the diagonal 1's allowance there.
Diagonal 3s
Two 3s touching only at a corner: the two outer sides of each — the ones away from the shared dot — are all lines.
Clue-Dot Balance
A clue and a nearby dot can force each other even when no second clue is needed.
Border 1-2-1
A 1-2-1 run on the border forces the middle border edge and rules out both 1s' inward and outer sides.
Three 3s in a Row
3-3-3 side by side: all four vertical edges crossing the run are lines.
Sector Chain
A sector is the pair of a cell's sides meeting at one of its corners, and the loop uses 0, 1, or 2 of them. A clue's opposite corners sum to it, so what travels is whether that count is odd or even — a 2 keeps it, a 1 or 3 flips it, and dots pass it on diagonally. A chain seeded at a border forces edges far from where it started.
Dead End
A line can never just stop — every dot touches zero or two line segments. Draw a candidate edge and follow where the line is forced to go: if it gets driven into a dot whose other edges are all crossed off, it dies in a dead end, so the candidate edge is ruled out.
Expert
Shared-Dot Balance
Several clues become linked because a dot fully enclosed by their cells must contain zero or two lines.
Long Dead End
Follow a candidate through at least eight forced continuations. If the chain eventually leaves the line with no exit, the original candidate is ruled out.
Crossing Parity
The loop is closed, so any straight line drawn across the whole board crosses it an even number of times. When every crossing edge in a row or column is decided but one, that last edge is forced — whatever makes the count even.
Sector Parity
Each cell has four sectors, one per corner, and the number of them holding exactly one line is always even. Once three of a cell's four sectors are pinned to "exactly one" or "never exactly one", the fourth is settled — even when its own two sides are both still blank.
Extreme
Bent Clue Trio
Three clues arranged around a bend can force edges only when their shared dots are checked together.
Extra Long Dead End
Follow a candidate through more than fifteen forced continuations. If that extended chain eventually leaves the line with no exit, the original candidate is ruled out.
Connectivity
The finished loop is a single connected shape, so every line already drawn has to end up joined to every other one. An edge marooned in a pocket the drawn lines can never reach is ruled out.
Region Coloring
Shading every cell as inside or outside the loop (with the space beyond the grid always outside): a line always separates two different shades, a blank always keeps the same shade. Chaining that logic across the board pins down edges too far apart for any single clue to reach alone.
Highlander
Uses the fact that the puzzle has exactly ONE solution: any move that would leave two different valid answers must be wrong. Powerful, but it's reasoning about the puzzle rather than about the loop.