Warps and wrap-around boards

A warp joins two cells on the edge of the board, so a path that walks off one of them comes straight back in at the other. A wrap-around board, sometimes called boundless, does this everywhere. The left edge joins the right edge and the top joins the bottom. Both take away the thing most solves start from, which is the edge.

The diagrams show a warp as a gap in the border with a short dotted line outside it. A path that uses a warp is drawn in two pieces, one ending at each warp cell. The solver at /solve/ doesn't read warps yet. We checked both boards on this page with the site's Python solver, which accepts warps in its JSON format, and each has exactly one solution.

What a warp changes

A warp gives each of its two cells one extra neighbor. An edge cell normally has three neighbors, and a warp makes it four, the same as a cell in the middle of the board. A corner cell with a warp gets three. That matters because edges and corners are where squeezes come from. A cell you'd normally treat as squeezed may have a way out through the warp, so check every warp cell before you call a move forced.

The other change is to walls. A path from one edge of the board to another normally splits it in two. With a warp, the two sides may still be connected through it, and a color that looks cut off can go around the long way.

A board with one warp

On this board, row 3, column 1 and row 3, column 5 are joined through the left and right edges.

Start with the corners. Orange's dot in the top-right corner has Red's dot beside it, so Orange's only exit is down. Blue's dot in the bottom-left corner has Red's dot beside it too, so Blue goes up. The corners without warps still work exactly as usual.

Find the wall. Orange runs from the top-right corner to its dot in the center, at row 3, column 3. Blue runs from the bottom-left corner to its dot at row 3, column 4, right beside Orange's. Whatever routes they take, the two paths together form an unbroken wall from one corner of the board to the opposite corner. Red's dot on the top edge is on one side of that wall, and Red's dot on the bottom edge is on the other.

On a plain board that would be a bottleneck with no way through. Here the warp is the only way past, so it has to be Red's, and neither Orange nor Blue can use it. Its left end is on Red's top side of the wall and its right end is on the bottom side. So Red runs along the top, down the left edge, through the warp, and down the right edge to the bottom row. Once Red is committed to that long route, Orange and Blue fill the middle and Green joins its two dots directly.

The warp also breaks checkerboard parity. Both ends of row 3 on a five-wide board are dark squares, so a step through the warp goes dark to dark. Without the warp this set of dots fails the parity count and has no solution. With it, the count no longer applies.

Wrap-around boards

On a wrap-around board every edge cell is joined to the cell on the opposite edge. There are no corners and no edges, and every cell has exactly four neighbors. The top-left cell touches the top-right cell and the bottom-left cell. Think of the board as a pattern that repeats in every direction. Draw it a second time beside itself and the paths continue across the join.

Follow the joins in the solution. Blue leaves row 1, column 5 upward and comes back in at row 5, column 5. Green leaves row 4, column 1 to the left and comes back in at row 4, column 5. Red's dot at row 5, column 1 is next to row 1, column 1, so Red reaches its partner by stepping off the top of the board.

What replaces the edge

Without edges, you need other things to start from.

Dots packed together. A dot surrounded by other dots is the one anchor a wrap-around board always keeps. On the board above, Red's dot at row 2, column 2 has Green's dots on both sides and Blue's dot below it. Its only exit is up, and that is the first move of the solve. Scan for dots touching two or three other dots, and remember to count neighbors across the joins. Green's dot at row 2, column 1 touches Blue's dot at row 2, column 5.

Finished paths. Every path you complete is a wall, and walls make new edges. On a wrap-around board, the first few paths you draw do the job the border does on a normal board, so finishing even a short pair early is worth more than usual.

Loops, not lines. A path from one edge to the opposite edge doesn't split a wrap-around board, because the board can be walked around the other way. To seal an area off, a wall has to close into a loop around it. Even a wall that runs all the way around the board, like a full row, leaves the rest connected the other way, and it takes two such rings to cut the board into separate bands. Check for that before you call a color cut off.

Expect to think ahead. After Red's first move the board above has no more forced moves, and most of what follows needs thinking ahead, the first step of it two moves deep. That's normal on wrap-around boards. Test moves near the most crowded group of dots first, since that's where a wrong move fails fastest.