Path puzzles: visit every square, then the ball

What a path puzzle is

Mathematicians call a route that visits every point exactly once a Hamiltonian path, after William Rowan Hamilton. In 1857 he invented the Icosian game, a puzzle about traveling round the corners of a twelve-sided solid, visiting each one once. Path puzzles on grids are the same idea, with squares in place of corners.

You’ll find the idea in lots of puzzles. The knight’s tour asks a chess knight to land on every square of a board once. Numbrix and Hidato have you fill a grid with numbers in a chain. LinkedIn’s Zip has you draw one path through every cell, passing numbered squares in order. Rex’s Fetch has no numbers: just a start, a finish, and a few things in the way.

How Rex’s Fetch works

• Throw the ball. Tap “Throw the ball!” and the Kid throws it from the porch into the yard. Where it lands is the last square of the route.
• Lead Rex. Drag him, tap the square next to him, or use the arrow keys. He moves up, down, left or right, never diagonally. The glowing squares show where he can go next.
• Every open square, once. Rex has to sniff every open square, and he can’t step on his own trail.
• Then the ball. He may step onto the ball only as the very last square.
• Bushes, flower beds and the birdbath block him. He can dig through dirt. Each yard has one of the family’s lost things buried somewhere along the way, and Rex digs it up as he passes.

A new yard comes every day at midnight, your time, the same for everyone. At the moment Monday to Saturday’s is 6 × 6, and Sunday’s is 7 × 7.

A worked example: the practice yard

Every new player’s first yard is a tiny practice one, 3 × 3. A bush fills the top-left corner. The ball is in the top middle. Rex starts in the bottom-right corner. That leaves six squares to visit between Rex’s first square and the ball.

You could try every route, but you don’t need to. Look at the corners.

1. The top-right corner has only two open neighbors: the ball, and the square below it. The ball comes last, so this corner must come second to last. The route ends: right-middle square, top-right corner, ball.
2. The bottom-left corner also has two open neighbors: the square above it and the square to its right. A square in the middle of the route needs a way in and a way out, so the trail must pass through both: left-middle, bottom-left, bottom-middle, in one order or the other.
3. Rex’s first step. He’s in the bottom-right corner, with two ways out: up, or left. Up leads to the right-middle square, which step 1 saved for near the end. So he goes left.

That’s the whole route: left along the bottom, up the left side, into the middle (where the dig spot is), right, up into the corner, and onto the ball. Three observations, and no guessing.

Bigger yards work the same way. You just make more of these small observations, and more often.

Tips for solving path puzzles

1. Start with the squares that have two ways out. A corner, or a square boxed in by bushes, often has only two open neighbors. The trail has to use both (unless it’s where Rex starts or the ball). Mark those pairs first.
2. Mind the ball. The ball is last, so the trail must arrive from one of its neighbors. If the ball has only one open neighbor, that square is second to last.
3. Watch for dead ends. If a square beside Rex has only one other way out, and it isn’t the ball, go there now, or it gets cut off.
4. Don’t split the yard. If your trail cuts the unvisited squares into two groups, Rex can only finish one of them. Before a long run, check that what’s left still hangs together.
5. Sweep in lanes. Long runs along an edge or a row of bushes are usually safe. The trouble comes from leaving a single square in a pocket that you’ll have to come back for.
6. Back up freely. Drag Rex back over his trail, or tap Back one. Backing up never counts against you.

For the curious: color the yard like a chessboard. Every step goes from one color to the other, so the number of light and dark squares has to fit the route’s length. You’ll rarely need it while playing, but it’s one of the checks the computer uses to rule yards out.

Why every yard has exactly one route

Each day’s yard is made from the date by a program, and then a route counter tries every trail Rex could take. The yard is kept only if exactly one route works. So there is always one answer, and you can reach it by reasoning like the example above.

Since October 23, 2026, the maker also draws many yards for each day and grades them by how much thinking they take: a step with only one sensible move is easy, and a fork with tempting wrong turns is hard. It keeps the yard closest to the day’s level. How we make puzzles with exactly one answer explains how.

Your result, par and the streak

Every yard has a par time, about what the fastest third of players take. It shows while you play. Finish under par for a gold paw. Your result shows your time and your streak, with a paw for each of the last seven days.

Share sends your time and what Rex dug up, never his route, with a link a friend can use to try to beat your time. Rex’s Fetch also has its own board, for your group of friends and for the whole world, fastest first.

More daily puzzles from the Fibbersons

Rex’s Fetch is one of three free dailies:

• Lie of Sight: a room seen from above, a family who each say what they saw, and one of them lying. About three minutes.
• Lacey’s Web: a number-and-connection puzzle on a spider’s web, in the family of Hashi bridges puzzles. See our Lacey’s Web guide.

All three are on the game picker, with the rules for each in How to play. For more grid logic, try a free daily logic puzzle or spatial reasoning puzzles.

Questions people ask