In Paths and Ribbons, a move can look perfectly legal and still make the board impossible. The trick is to look at the space a route leaves behind, not just where it’s heading.

Paths has one route through numbered checkpoints. Ribbons has several pairs to connect. Both need every cell filled exactly once. The boards below are small teaching examples. In Ribbons, each pair has its own shape as well as its own colour, so circles and triangles tell the pairs apart in case the colours are hard to distinguish.

Trap 1: joining a pair too soon

On this Ribbons board, the two circles sit side by side along the top, and joining them directly looks like the obvious move. But look at the top-right corner. It isn’t an endpoint, so a ribbon has to enter it from one side and leave by another. If the circles join directly, the corner is left with only one free neighbour, the cell below it. No ribbon can pass through it, and the triangle ribbon can’t use it either.

Three-by-three Ribbons board. A ribbon joins the circle endpoints in the top-left and top-middle cells directly. The outlined top-right corner has only its lower neighbour free. The triangle endpoints are in the two bottom corners.

A finished pair can strand an empty cell

Don’t take the short connection. If the circle ribbon heads down first instead, it has room to loop round through the top-right corner and back to its partner.

Trap 2: reaching the finish too early

On the Paths board, a wall under checkpoint 1 means the route has to start by going right. After reaching 2 in the top-right corner, it’s tempting to head straight down to 3. The order is right, but four cells on the left haven’t been visited. Because 3 is the last checkpoint, the route ends there, and those four cells are lost.

Three-by-three Paths board. The route goes from 1 across the top to 2 and down the right edge to final checkpoint 3. Four cells at the lower left remain unvisited.123

Order isn’t the only rule

The final checkpoint has to come last, after every other cell. From the middle-right cell, turn left into the centre instead of heading down.

Trap 3: splitting what’s left

Now take that turn into the centre, but then go down. The route ends at the bottom-middle cell, and what’s left has split into two groups: two cells on the left, and the final checkpoint on the right. Both can only be reached through the end of the route. Go left and you can’t get back to the finish. Go right and you finish with the left cells still empty. A single route can’t do both.

The Paths route reaches the centre then turns to the lower-middle cell. Two outlined left cells and the final checkpoint on the right form separate branches from the current path end.123

Go back to the turn that caused the split

Return to the centre and turn left. Visit the middle-left and bottom-left cells, then cross the bottom row to the finish.

The repaired routes

Both solutions below fill every cell with no overlaps and no repeats. The Paths route visits 1, 2 and 3 in order and ends on 3. In Ribbons, the circle ribbon takes the long way round through the middle and the top-right corner, leaving the bottom row for the triangles. The longer route is the only one that leaves every cell coverable.

Complete Paths solution: top row from 1 to 2, back across the middle row, then across the bottom row to 3.123
Paths: one route through all nine cells.
Complete Ribbons solution. The circle ribbon travels down from the top-left, across the middle, up to the top-right and left to the top-middle. The triangle ribbon connects the bottom corners across the bottom row.
Ribbons: both pairs joined, all nine cells covered.

Four questions before each turn

  • Does every empty corner still have a way in and a way out?
  • Can every unfinished pair or checkpoint still be reached?
  • Is what’s left still one connected space, or has this turn split it?
  • Am I about to finish while cells are still empty?

If a move fails one of these, work out exactly what’s blocking it before you redraw. A reason you can name is one you’ll spot sooner next time. For a more general reset, see what to do when you’re stuck.