The chequerboard trick for Paths

When two moves on a Paths board both look fine, I sometimes shade the board like a chessboard and count squares. It takes a few seconds, and now and then it settles the question outright. Here’s how it works, and when it won’t.

First, the rules. Your route starts at checkpoint 1, passes the numbered checkpoints in order, visits every square exactly once and finishes on the last checkpoint. It moves up, down, left or right, never diagonally and never through a wall.

Every step changes colour

Shade the squares like a chessboard, corners included. A step in any of the four directions lands on the other kind of square, so the route runs shaded, plain, shaded, plain, all the way along. Two things follow.

The first is about checkpoints. If two checkpoints are both shaded, or both plain, the route takes an even number of steps from one to the other. If one is shaded and the other plain, it takes an odd number.

The second is about the whole board. A 5×5 board has 13 shaded squares and 12 plain ones. A route that alternates through all 25 has to start and finish on shaded squares, the kind there’s one more of. On a 4×4 or 6×6 board the two kinds are equal, so the route starts on one and finishes on the other.

Every published board already obeys both, so neither will catch one out. Their use is the next idea.

Count an area, not just a line

Any stretch of the route alternates too. So when you know the route has to fill a particular area, count its shaded and plain squares. A visit that goes in and comes out on shaded squares covers one more shaded square than plain. In and out on plain squares, one more plain. In on one kind and out on the other, the same number of each.

Try it

This 5×5 board was made for this note; it isn’t from the Library, the Daily set or the Weekly collection. Rows are counted from the top and columns from the left. There are two walls: one under the middle square, and one under the right-hand end of row 3.

Up is tempting: checkpoint 2 is in the middle, and the left edge looks like a tidy way there. The usual dead-end checks don’t object either. After going up, every empty square still has at least two open neighbours, and the empty squares are all still connected.

All five checkpoints are shaded, so every stretch between them takes an even number of steps. True, but no help with this choice. Counting an area is.

Look at the bottom-right block: rows 4 and 5, columns 2 to 5. Like any block two squares deep, it has four shaded squares and four plain ones. The walls close its top in columns 3 and 5 and the board’s edges close the rest, so a route can enter it by only three squares. Row 5, column 2 is plain, and from outside it can only be reached from checkpoint 1. Row 4, columns 2 and 4 are both shaded.

Now suppose you go up. The plain way in has gone, so every visit to the block would have to start and finish on shaded squares, and each of those visits covers one more shaded square than plain. One visit or three, you’d use more shaded squares than plain ones, but the block has four of each. The route can’t fill it, so going up is wrong. Go right.

The count tells you a little more. Going right enters the block on a plain square, so this visit has to leave from a shaded one, which keeps it balanced. Any later visit would be shaded at both ends and spoil the balance, so there isn’t one: the route fills all eight squares before it leaves. The colours can’t tell you which shaded square it leaves from; the rest of the board decides that.

When it helps, and when it doesn’t

The trick earns its keep when an area has only a few ways in, hemmed in by walls, edges or route you’ve already drawn, and is small enough to count at a glance. It’s good at telling you which way in to use, or that an area can’t be left until later.

Otherwise it stays quiet. Any route you can actually draw between two squares already has the right odd or even length, so the checkpoint fact never rules out a move by itself. The colours can’t choose between two ways in of the same kind, and they say nothing about checkpoint order. Dead ends and moves that cut the board in two are quicker to spot, so check for those first and reach for the shading when they run out.

Try it on a Paths board in the Library, or read the Paths guide for the rules and controls. The quicker checks mentioned above are in avoid route traps.