One way across, and only one
Step from the start to the finish through touching cells, never using one twice, collecting exactly the target. Every board is checked by exhaustive search: precisely one route does it, and nothing else on the board comes to the same total.
Why the route is always fairly direct
Because a wandering one cannot be the only answer, and we would rather say so than pretend otherwise.
The obvious way to build a puzzle like this is to draw a long looping route, fill the grid at random, and publish the route's total. That board is broken and does not look it. On a 6 × 6 grid a route of eighteen cells has a total that roughly twenty thousand other routes also reach — measured, not estimated. A player finds one of them, it adds up perfectly, and the puzzle tells them they are wrong.
What can be proven is a target near the cheapest way across. Every value on the board is at least one, so a detour only ever adds; the cheapest route is therefore also a shortest one, and totals just above that floor are reached by very few routes indeed. Scanning upward from the floor finds totals that exactly one route hits, and an exhaustive count confirms it.
So the answer here is short and reasonably direct. You are still free to double back — the guarantee is over every route on the board, not only the direct ones — but you will find that doubling back overshoots.
What the difficulty actually measures
Not route length, which barely varies: on a 6 × 6 the answer is eleven cells in nearly every board, in every band. Measuring difficulty by it would produce three labels that all mean the same thing.
What differs is how close the wrong routes get. You solve one of these by adding as you go and backing out when the total overshoots, so the routes that matter are the ones that survive that all the way to the finish and are still wrong. Every board is scored by how many routes land within two of its target — Gentle at most four, Steady up to eighteen, Twisty more than that. On a twisty board being close tells you almost nothing.
How to play
Click or tap the start, then any cell touching the last one you took. Click the last cell again, or press Step back, to undo. Arrow keys move a cursor and Enter takes the cell; Backspace steps back.
The running total sits above the board with the gap to the target beside it, because the arithmetic is the puzzle and doing it twice in your head is not the interesting part. Overshooting is not a failure state — it just means the route is dead, and the fastest way forward is usually to back out two or three cells rather than start again.
Where to start
Work out roughly how many cells you can afford. The target divided by the average value on the board tells you whether you are looking for a direct route or one with a detour in it, and that alone rules out most of the board.
Then work from both ends. The finish usually has only two ways in, and one of them is often far too expensive to be part of a route that hits the target exactly. Ruling that one out is worth more than any number of forward guesses, which is also what the second hint gives you.
Related
Number Forge for reaching a target from a handful of tiles, Arithmetic Cages for arithmetic under a Latin-square constraint, or the puzzle hub for the rest.
Questions people actually ask
Can two different routes reach the target?
No. Every board is handed to an exhaustive search that enumerates routes and stops once it has found two; it is published only when the count is exactly one. That check runs against the grid as you receive it, not against anything the generator remembers about how it was built.
Why is the answer never a long looping route?
Because a long route cannot be the only one reaching its total. On a 6 × 6 board a route of eighteen cells shares its total with around twenty thousand others — we measured it. Only totals near the cheapest way across are reached by few enough routes to be provably unique, and those routes are short. You can still double back; the guarantee covers every route on the board, not just the direct ones. You will just find that doubling back overshoots.
What does the difficulty change?
How close the wrong routes get. Route length barely varies here, so labelling by it would give three names for the same thing. Instead every board is scored by how many routes land within two of the target: at most four for Gentle, up to eighteen for Steady, more for Twisty. On a twisty board, being close tells you almost nothing.
Why is every number at least one?
Because the generator relies on it. With all values positive, a detour can only add to a total, which makes the cheapest route also a shortest one — and that is what makes the whole thing provable in milliseconds rather than minutes. A single zero on the board would break the argument.
Are hints just the answer?
Only if you ask three times. The first press says which step is in question and how many ways lead on from it. The second rules a cell out rather than naming the right one, which is usually worth more — a wrong step early costs a lot of retracing. Only the third gives the move. If your route has already left the only correct one, it says so instead.
Is my progress saved?
In your browser, on this device. There is no account and nothing is sent anywhere. Clearing your browser data clears it.