Bottlenecks and cut-off colors
Learn to see narrow passages, moves that cut a color off from its partner, and gaps too small for the colors that need them.
A dead end leaves a cell that nothing can fill. A bottleneck leaves a color that can't finish. Either its two ends no longer have a route of empty cells between them, or one of its tips has no open side at all. Both mistakes are easy to make, because the move that causes them usually looks like progress for some other color.
The fix is a habit. Before a move, check that every color near it can still reach its partner.
Cutting a color off
A color is cut off when every route between its two ends is blocked by other paths, dots, holes or the edge of the board. It doesn't matter how much empty space is left. If none of it connects the two ends, that color is finished.
Blue's dots are at row 1, column 1 and row 3, column 1, with Orange's dot squeezed between them. Blue had to start along the top edge, and Orange has stepped right to row 2, column 2. Blue's partner is close, so turning down at row 2, column 3 looks like the start of a short loop home.
Check Orange before you draw it. Orange's tip at row 2, column 2 would have one open side left, row 3, column 2. That cell is surrounded by Blue's dot, a Green dot below and a Green dot to the right. Orange's other dot is at row 4, column 3, and there would be no way to get there. Orange needs row 2 to travel east, so Blue can't take it.
The same thing happens at the next two chances Blue gets. Turning left at row 2, column 4 or at row 3, column 4 also closes Orange's lane. So Blue runs along the whole outside of the board, and Orange gets the inside.
This puzzle shows the most common bottleneck pattern. A long path wants to take a shortcut through the middle, and the shortcut crosses the only lane another color has. When a path seems to be taking the long way round, look for the color it is making room for.
You can skip some colors when you check. A color whose two ends are already side by side can never be cut off, since it can always join where it stands. The colors at risk are the ones whose ends are far apart and whose only routes run past the move you're about to make.
Boxing in a path tip
The quicker version of the same mistake is stepping into a cell with no way out. The tip then has no open neighbor, and unless its partner is right next to it, the color is stuck.
Blue has gone around the corner from its dot at row 1, column 2 and down to row 3, column 1. Its partner is at row 4, column 3, so stepping right toward it is tempting. But row 3, column 2 is next to Orange's dot above, a Red dot to the right and the other Red dot below. Blue would arrive and find no empty cell and no Blue end. Blue goes down two and right two instead, and the cell Blue wanted ends up joining the two Red dots.
Boxing in is easy to check. Before a tip steps into a cell, look at that cell's other sides. If they are all full and none of them is the same color's other end, the step is wrong. Cells surrounded by several dots are the usual traps.
Count the colors, then count the gap
Bottlenecks get more interesting when several colors need the same narrow place. Each path crossing a gap uses up at least one of its cells, and paths can't share cells. So if more colors need to cross a gap than it has cells, the position is broken, even if every individual color seems to have a route.
The holes in column 3 split the board into two halves, joined only at row 3, column 3. Blue has a dot on each side, and so does Orange. Taken one at a time, each color can reach its partner through the gap. Together they can't, because only one of them can have that cell.
Open one more cell in the wall and the count works.
Now there are two crossing colors and two gap cells. In the solution, Blue crosses at row 3 and Orange at row 4. When the numbers match exactly like this, every gap cell has to go to a crossing color, and that often forces moves on both sides.
Holes make the wall easy to see, but drawn paths make the same kind of wall. In a real position, look for a narrow place between finished paths, dots and the edge. Imagine a line across it. Count the colors that have one end on each side of the line, then count the open cells on the line. If the first number is bigger, something already drawn is wrong. If the numbers are equal, the gap is fully booked. A path that only wants to pass along the gap, without crossing, has no room there.
Splitting the board into regions
The same counting works on a larger scale. When a path runs from one edge to another, or curls round to meet other paths, the empty cells can split into separate regions. Then two quick checks apply:
- Every unfinished color must have both ends touching the same region. A color whose ends border different regions is cut off.
- Every region needs a color that touches it with both ends. Otherwise the region is a sealed pocket, as described in dead ends and stranded cells.
These checks also make a split useful. Once you know which colors belong to which region, each region becomes a smaller puzzle you can solve on its own. On a big board, finding the move that splits it cleanly is often the key step. Solving large boards covers that in more detail.
Making it a habit
Before you commit a move, look at the colors whose routes run through the area the move touches:
- Does any tip next to the move lose its last open side?
- Does any color's only route pass through the cell you're about to fill?
- Does the move narrow a gap below the number of colors that need it?
The first two catch most mistakes. The third catches the ones that cost the most time, because a gap can be overbooked long before any single color looks stuck.
Sometimes a move only creates a bottleneck after a chain of forced replies. Following that chain is covered in thinking ahead. For practice, the bottlenecks practice set lists puzzles whose walkthroughs include a cut-off or boxed-in color.