Hexagonal boards

On a hex board every cell has up to six neighbors instead of four. A path tip almost never has a single exit, so the chains of forced moves that carry a square board are shorter and start later. What still works, and works well, is looking at empty cells near the corners and asking who can fill them.

The solver at /solve/ doesn't read hex boards yet. The boards on this page were checked with the site's Python solver, which supports them.

Reading the layout

The diagrams use an offset layout. Rows 1, 3, 5 and so on sit flush left, and rows 2, 4, 6 are shifted half a cell to the right. Rows and columns are still counted from the top left, so you can name cells the same way as on a square board.

Each cell touches the two cells beside it in its row, two cells in the row above and two in the row below. Which two depends on the row:

In a flush-left row (1, 3, 5), cell c touches columns c − 1 and c in the rows above and below.
In a shifted row (2, 4), cell c touches columns c and c + 1 in the rows above and below.

Corners and edges

On a square board every corner has two neighbors. On a hex board it depends on the corner's shape. Where the outline makes a sharp point, the corner has two neighbors and behaves like a square corner. Where it makes a blunt one, the corner has three.

On the boards here, the corners of the flush-left rows on the left side are sharp. Row 1, column 1 touches only row 1, column 2 and row 2, column 1. The corners on the right of those rows are blunt, with three neighbors. With an even number of rows the bottom row is shifted, and the pattern flips there. On a board shaped like one big hexagon, all six corners are blunt, so there are no two-neighbor cells to start from at all.

Edges are uneven too. Along the left side, cells in flush-left rows have three neighbors and cells in shifted rows have five. Along the top and bottom, edge cells have four.

Why forced moves are rarer

On a square board a dot with two neighbors taken has two exits left. On a hex board it still has four. It takes a crowd of dots and paths to cut a tip down to one exit.

No dot here has only one exit. The first deduction comes from the empty cell at row 4, column 5.

Check the dots on this board. Blue's top dot has three exits, Green's right-hand dot has five, and Orange's center dot has six. Orange's dot in the bottom-right corner and Green's bottom dot have two each. There's nothing to draw yet.

Squeezes still work

A squeeze is about an empty cell, and empty cells near corners and dots still run short of neighbors. That's where a hex solve starts.

The first move. Look at row 4, column 5. It touches three cells: row 4, column 4, Green's dot above it at row 3, column 5, and Orange's dot below it at row 5, column 5. Orange's corner dot has two exits, this cell and row 4, column 4. If Orange goes to row 4, column 4, this cell is left touching only Orange's new tip and Green's dot, and whichever color steps into it is boxed in. That's a dead end, so Orange goes up into row 4, column 5.

The chain that follows. From there Orange's only exit is row 4, column 4. That leaves Green's bottom dot one exit, row 4, column 3, so Green goes there. Orange then has one exit, row 3, column 4, which touches its partner, and Orange is done. Green's tip at row 4, column 3 is now hemmed in by Orange and Blue and can only go to row 4, column 2. Blue's bottom dot has lost every neighbor but row 5, column 2, so it steps left.

Orange is finished. The sharp bottom-left corner now has two usable neighbors, and one is Blue's tip.

The sharp corners. Row 5, column 1 touches only row 5, column 2 and row 4, column 1. Blue's tip is one of them, so Blue runs through the corner and up to row 4, column 1. From there Blue climbs the left edge, and Green, squeezed between Blue and Orange, climbs beside it to row 2, column 2. Then the top-left corner does the same thing: it touches only row 1, column 2 and Blue's tip at row 2, column 1. Blue turns through it and runs along the top to its partner.

Green finishes along row 2. The last cell, row 2, column 5, is an edge cell with three neighbors. One is Blue's finished dot, and the other two are Green's tip and Green's dot, so Green passes through it and the board is full.

The solution. Blue follows the left edge through both sharp corners, and Green wraps around Orange.

What carries over and what doesn't

Dead ends carry over unchanged. An empty cell that can't get two usable neighbors, or one trapped between tips of two different colors, kills the position on any grid. On hex boards this is often the first deduction, as it was here.

Bottlenecks carry over. A path that runs from one edge to another still splits the board in two, and a color with dots on both sides is cut off. Hex paths can bend in six directions, so walls tend to be more ragged, but the test is the same.

Forced runs are shorter. Once a region gets crowded, forced moves come in runs, as Blue's did along the left edge. They stop sooner than on a square board, because each new cell usually has a fresh spare neighbor.

Checkerboard parity doesn't apply. Any three cells like row 1, column 1, row 1, column 2 and row 2, column 1 all touch each other, so the cells can't be shaded in two alternating colors. The parity count has nothing to say about hex boards.