Grid Paint

A cursor sits somewhere on a small board and leaves a trail behind it. Step it up, down, left or right until every open square has been painted — once each, never twice, and never lifting the trail to start again somewhere else. Mathematicians call the thing you are looking for a Hamiltonian path; the version here is cut down to a six-by-four board with three squares walled off, so twenty-one are left to paint — small enough to finish in a couple of minutes, and shaped so that the shape rather than the size is what makes it hard.

The walls are why it is a puzzle at all. A bare rectangle entered at a corner is finished by sweeping along one row, dropping down and sweeping back, which teaches nothing. Put three walls in and the board grows pinch points and pockets, and the question becomes which squares you can afford to leave until later. Every board is dealt only after a search has found a complete route through it from your starting square, so a dead end is always the route you chose and never the board you were given.

It runs entirely on your device: 0 network requests, works offline. No ads, no account, nothing to buy, and no betting, money or prizes of any kind.

1. Focus the board — click it, or press Tab until it is outlined.
2. Step with the arrow keys, or W A S D. The cursor is marked ◉ and moves one square at a time. A wall, the edge of the board and any square already painted are all refused, so nothing is ever lost by trying a direction.
3. Keep going with Enter or Space — it repeats whichever way you last moved, which is the quick way along a corridor.
4. Take the forced step with F. When exactly one square is legal there is nothing to decide, and this saves you finding which one it is.
5. Or just tap. Clicking any square next to the cursor steps onto it.
6. Read the small numbers. Each square you could step to right now is marked ◇ with the number of ways out you would still have from there. Squares further off are marked △ when they are down to a single way out, and × when they have none left at all.
7. Watch the loose-end count in the status row. A square with one way out has to be where the trail finishes, and a trail has only one finish — so one loose end is fine and a second means the board is already lost, however many squares are still blank.
8. Do not seal yourself in. If the cursor has nowhere legal to go while squares are still unpainted, the round ends there and the squares you never reached are numbered with a route that would have worked.

How do you visit every square of a grid exactly once?

By working on the squares you are about to lose rather than the ones in front of you. The rule that decides almost every board is a counting one: everything still to paint has to form a single line, and every square in the middle of a line is entered once and left once, so it needs two ways out. A square down to one way out can therefore only be the very end of the trail — and there is only one end. So the moment a second square is down to one way out, the board is finished whether or not it looks it, and the moment a square is down to none it is gone. That is why this game prints the number of ways out on every square instead of leaving you to count them: the arithmetic is not the interesting part, the decision is. The practical version of the rule is to clear the awkward corners and pockets early, while they still have two ways in, and leave the well-connected middle for last, because the middle stays reachable long after the edges have quietly closed.

Why can the trail only start on certain squares?

Because of a chessboard argument you can check in your head. Colour the board light and dark like a chessboard; every step up, down, left or right lands on the opposite colour to the one it left. So a trail through twenty-one squares alternates all the way and uses eleven of whatever colour it began on and ten of the other. If the shape has eleven dark squares and ten light, the trail must begin on a dark one — start on a light square and you would need eleven light squares, which do not exist. Handing you a start like that would be a puzzle nobody could solve and nobody could tell was unsolvable, so the generator never offers one. The same counting also throws out whole wall layouts before any searching happens: if the two colours differ by more than one, no square on that board can begin a full trail.

What happens when I get stuck?

The round ends at once and counts as a loss, and the squares you never reached are numbered with one complete route from your starting square — so you can see where the trail should have gone. Ending it there is deliberate. Leaving a player to wander a board that can no longer be finished is worse than telling them, and the move that killed it is still in mind at the moment it happens rather than ten steps later. There is no undo either: a trail you can rub out is a different and much easier puzzle, and the whole lesson is in committing. Squares painted still count towards the score, so a trail that dies on square eighteen reads like the real progress it was.

Is there any money, betting or in-app purchase in this?

None, and there never will be. There is no currency, nothing to buy, no lives to refill and nothing to win or lose but the puzzle. The score is derived from your finished board, and it stays on your device.

Does it work offline?

Yes. After the page has loaded once, the board, the rules and the route search all live on your device, and nothing is sent anywhere while you play. It is the same code with the network on or off.

Can I play it with a keyboard or a screen reader?

Both. Arrows or W A S D step, Enter repeats the last direction, F takes the only legal step, and every square announces itself in words — "cursor here, step 7", "painted, step 3", "legal step, 2 ways out from there", "blocked". Nothing depends on telling colours apart: the cursor, a painted square, a legal step, a square with one way out and a wall each carry their own glyph.

Twenty-four squares with three of them walled, one deal at a time, no undo and no timer. The trail is open, so it does not have to finish anywhere in particular — only cover everything. The "gentle", "steady" and "tight" rating is how many complete routes the board has from your start, counted up to twenty-four; above that it simply reads gentle rather than counting further. The ways-out numbers are a fact about the position, not a hint about the answer: they tell you what is true right now, and there is no board where following them blindly is guaranteed to work.

Knight Hop is the same problem with a different step — every square once again, but the piece moves like a chess knight, so the pockets appear in places that look nothing like pockets. Tilt Maze is the opposite temperament: you cannot choose where to stop, only which wall will stop you.

Bookmark this page (Ctrl+D, or ⌘D on Mac) or install the app — it works offline the next time you need it.