Shaped boards and holes

A hole is a cell that isn't part of the board. Nothing can enter it and nothing needs to fill it. Cut a few holes into a rectangle and you get a notched outline, a wall down the middle or a frame. Holes often make a board easier, because every hole takes neighbors away from the cells around it.

Any outline you can draw on a square grid is a rectangle with holes in it, and the solver at /solve/ accepts holes entered by hand. The diagrams draw holes as gaps in the grid.

Holes make new corners

A corner is just a cell with two neighbors. On a plain rectangle there are four of them. A notch in the outline adds more, and they work exactly like the real ones. If one of a corner's two neighbors is a dot or a path tip, that color has to run through the corner.

Look for corners in two places:

• Inside a step. Where the outline steps in, the cell tucked into the step loses two neighbors to holes.
• Beside a single hole on the edge. An edge cell has three neighbors. A hole next to it takes one more.

Cells beside holes in the middle of the board lose a neighbor too. They become edge cells with three neighbors, which makes dead ends easier to create.

This board has the top-left and bottom-left corners cut away in steps. Count the neighbors of each marked cell. Row 1, column 3 touches only row 1, column 4 and row 2, column 3, and the other three marked cells are the same. Blue's dot at row 2, column 2 is in a corner of its own, with only row 2, column 3 and row 3, column 2 beside it.

The solve uses those corners almost at once:

• Blue at row 1, column 5 is boxed in by Orange's two dots, so it moves left. At row 1, column 4 Green's dot blocks the way down, so Blue continues into the new corner at row 1, column 3, which turns it down to row 2, column 3, right beside its partner.
• Red at row 4, column 1 has a hole below it and Green's dot beside it, so it goes up into the corner at row 3, column 1. From there Red is squeezed between Blue's and Green's dots and runs along row 3, then down to its partner at row 4, column 4.
• Green ends up wrapping the whole bottom of the board. Its route from row 4, column 2 runs through the corners at row 5, column 2 and row 6, column 3, because each of them has Green's dot or tip as one of its only two neighbors.

Holes make bottlenecks

A line of holes with a gap in it splits the board into rooms joined by a doorway. A doorway one cell wide is a cell with two neighbors, one in each room, so whatever fills it has to pass straight through from one room to the other. That makes it a bottleneck you can read before drawing anything. Exactly one color goes through, and it has to be a color with dots in both rooms.

List the dots in each room. The left room has Blue's two dots, Orange's two dots and one Red dot. The right room has Green's two dots and the other Red dot. Red is the only color split between the rooms, so Red takes row 3, column 4, and no other color may enter it.

The rest follows quickly. In the left room, Blue's dot at row 5, column 3 has the wall on one side and Red's dot above it, so it goes left and up. Orange follows the left edge and the top row. Red climbs from row 4, column 3 through the doorway. In the right room, Green's dot at row 5, column 5 has the wall beside it and Red's dot to its right, so it goes up, and Red takes the outer cells of the room on the way to its partner.

Two things to check on boards with walls like this:

• Count the doorway. If a gap is two cells wide, up to two colors can cross. Count the colors that have to cross and compare. More colors than cells means a mistake.
• Recount parity. Holes change how many dark and light cells the board has, so the checkerboard count can have a different target on a holed board. Recount before you use it.

Rings, cubes and linked loops

Some boards aren't square grids at all. The site's Python solver models several of these, but the browser solver doesn't read them yet.

Rings. Cells sit in circles around a center, like the sections of a target. Each cell touches the two cells beside it in its ring, which closes on itself, and the cells directly inside and outside it. There are no corners at all. Cells on the inner and outer rims have three neighbors and the rest have four. Some ring boards fill the center with a single core cell that touches every cell of the innermost ring. Squeezes only appear once dots and paths start taking neighbors away, usually along a rim.

Cubes. Three square faces are drawn as one corner of a cube. Cells along the seam between two faces touch each other across the seam, so a seam isn't an edge even though it looks like one on screen. The three cells at the point where the faces meet each touch a cell on both of the other faces. Only the outer outline is a real edge.

Linked loops. The board is a track of irregular cells that closes into loops, and the loops meet at a few places, as in a figure eight. Many cells on a track have only two or three neighbors, so forced moves can run a long way. The places where loops meet are doorways between them, and the doorway count from above tells you which colors have to pass through.

On all of these, settle first which cells actually touch. Go by the board's lines. Two cells that look close on screen may not touch at all.