Sudoku Made and maintained by Brian Hamilton and Karan Hamilton.
Brian builds the playable puzzle tools, while Karan helps test clarity, difficulty feel, and the player experience.
Playable puzzles and controls are tested in the browser for rule accuracy, usability, and a clear solving experience. See how we make and test puzzles, our editorial standards, or report a problem.
What is Arrow Sudoku?
Arrow Sudoku is ordinary sudoku with arrows drawn over the grid, each starting in a circled cell. Fill it so that:
- every row, column and 3×3 box holds 1 to 9 exactly once;
- and the digit in each circle equals the sum of the digits along its arrow, from the cell next to the circle to the arrowhead.
The circle is the total, not part of the sum. A path is two to five cells long on this page, most often two or three, and may turn diagonally; arrows never share a cell, and a digit may appear twice on one path as long as the two cells are not in the same row, column or box. Because the circle holds a single digit, no path ever adds up to more than 9.
If you know Killer Sudoku, the arrow looks like a cage with its total written in a cell. Two things differ. A cage never repeats a digit; an arrow may, provided the repeats sit in different houses, so 1 + 1 + 2 is a legal three-cell path when its two 1s see nothing of each other. And the total is itself a cell in the grid: the circle’s digit answers to its own row, column and box, so a 7 in the circle’s box rules 7 out of the circle, and with it every path that adds up to 7.
What does an arrow tell you before you place anything?
Read the arrows first. A path of n cells adds up to at least n, so its circle is at least n and every path cell is at most 9 − (n − 1); a three-cell path never holds an 8 or a 9. Then every digit already on the arrow tightens the rest: a given circle fixes what the path must add up to, and a given path cell takes its digit out of the circle’s budget. Write what survives in as pencil marks, and note the sets that fit:
| Circle | Two-cell path | Three-cell path | With a repeated digit |
|---|---|---|---|
| 2 | — | — | 1+1 |
| 3 | 1+2 | — | 1+1+1 |
| 4 | 1+3 | — | 2+2 1+1+2 |
| 5 | 1+4, 2+3 | — | 1+1+3 1+2+2 |
| 6 | 1+5, 2+4 | 1+2+3 | 3+3 1+1+4 2+2+2 |
| 7 | 1+6, 2+5, 3+4 | 1+2+4 | 1+1+5 1+3+3 2+2+3 |
| 8 | 1+7, 2+6, 3+5 | 1+2+5, 1+3+4 | 4+4 1+1+6 2+2+4 2+3+3 |
| 9 | 1+8, 2+7, 3+6, 4+5 | 1+2+6, 1+3+5, 2+3+4 | 1+1+7 1+4+4 2+2+5 3+3+3 |
Every set of digits that adds up to the circle, in any order along the path. The last column repeats a digit, which is legal on an arrow only when the repeated cells share no row, column or box; on this page 70% of Easy paths and a little under half of Hard and Expert paths sit inside one house, where those sets are out. A four-cell path cannot fit inside one house at all: 1 + 2 + 3 + 4 is already 10.
That first read does more than it looks. Across 40 boards per level it strikes a median of 16 pencil marks on Easy and 46 on Expert before any digit is placed, and settles four to six cells outright. Cells on an arrow start no tighter than the rest of the grid — 3.4 candidates each on Easy against 3.3 elsewhere — and finish the read at 1.8 to 2.7, roughly half. On Hard and Expert, where every path is three cells or more, the circle is a 6 or above nine times in ten.
Do the arrows actually do any work?
Every board this page deals is built so that the answer depends on them: the generator removes digits until the puzzle has one answer under arrow rules, then checks the same givens under plain sudoku rules and keeps digging until they no longer have one. On the board above:
We checked 160 fresh boards, 40 per level: every one has a single answer with its arrows and more than one without them. Without them, Easy givens have a median of eleven answers and up to 77, Medium a median of 46, and Hard and Expert more than 200 on every board we tried. A solver that ignores the arrows stalls on every board at every level.
How do the four levels differ?
We ran 40 fresh boards per level through a solver that knows one technique at a time, then took techniques away to see which boards stopped:
| Level | Given digits | Arrows | Cells on an arrow | Marks the first read strikes | Reading and singles finish it | Needs locked candidates or a subset | Beyond our solver |
|---|---|---|---|---|---|---|---|
| Easy | 28–32 | 4–6, paths of 2–3 cells | 16 | 16 | 100% | 0% | 0% |
| Medium | 28 | 4–6, paths of 2–4 cells | 16 | 18 | 90% | 3% | 8% |
| Hard | 25 | 5–7, paths of 3–5 cells | 25 | 33 | 93% | 0% | 8% |
| Expert | 21 | 6–8, paths of 3–5 cells | 29 | 46 | 80% | 10% | 10% |
40 boards per level from this page’s generator, fixed seed; medians, and shares rounded. The solver reads every arrow to a fixed point, then knows naked and hidden singles, locked candidates, naked pairs and triples, and the X-Wing; “beyond our solver” boards still have exactly one answer. Easy comes up to four givens short of its target when that is what it takes to make the arrows load-bearing. Five-cell paths are allowed on Hard and Expert and rare, about one arrow in 250, because a five-cell sum of 9 or less needs repeated 1s and 2s that share no house.
Up to Hard, the levels differ in how much you are told; Expert is where the techniques change. Arrow reading plus naked and hidden singles finish nine boards in ten on Easy, Medium and Hard. On Expert that falls to eight in ten: one board in ten needs locked candidates or a naked pair or triple, and one in ten goes past everything our solver knows. Expert also has the most reading to do — 21 givens, six to eight arrows over 29 of the 81 cells, and 46 marks struck by the first read alone. This page used to say Hard already demanded advanced techniques; we measured, and Hard is singles more than nine times in ten.
The generator takes a finished grid, draws the arrows along it, and removes digits one at a time, keeping a removal only while the puzzle still has exactly one answer under arrow rules; a search it cannot finish answers “not proven” and the clue stays. We re-solved 160 fresh boards with a separate counter written for the purpose: every one had exactly one answer, at its level’s clue target. A board takes about 2 milliseconds to build, 9 at worst, so it is ready before the page has finished loading.
Which level should you play?
Medium is where this variant is at its best. Twenty-eight givens and four to six short arrows: the reading does real work on every board — the same givens have 46 answers without it — and nine boards in ten fall to reading and singles. Easy is the same arrows with more givens and two-cell paths, and the right place to learn what a circle promises.
Hard is more arrows and longer ones, not a new skill: three-cell paths, nearly a third of the grid under arrows, and still singles more than nine times in ten. Expert is the real step. One board in ten needs locked candidates or a subset and one in ten needs something beyond them, more than any other level on this page. Every one still has exactly one answer, and none needs guessing, but an Expert board that feels impossible rather than hard is usually one of those.
What goes wrong on an Arrow Sudoku
- Adding the circle into its own sum. The circle is the answer; only the cells beyond it are added.
- Treating the arrow as a cage. Digits may repeat on a path when the repeated cells share no house. A three-cell path adding up to 5 is not impossible; it is 1 + 1 + 3 or 1 + 2 + 2 with the repeats kept apart.
- Putting a 9 on a path. It cannot be done: the rest of the path adds at least 1, and the circle cannot hold 10. An 8 fits only on a two-cell path beside a 1.
- Reading an arrow once. Every digit placed on it, or in a house that meets it, changes what the rest can be. Re-read the arrow each time; the second read is where most placements come from.
- Leaving the arrows for later. Ignore them and every board on this page has several answers, so a solve that contradicts itself late usually went past an arrow without reading it.
Where does Arrow Sudoku come from?
Arrow Sudoku is a Japanese number-place variant — nanpure is the Japanese name for the puzzle family — and by the account on Thomas Snyder’s puzzle site it was likely in print there before sudoku made its return to the West in 2004. It reached competition solvers through events like the 2010 Hungarian Sudoku Championship, for which Snyder set an arrow puzzle, and the YouTube channel Cracking the Cryptic has kept it in front of a wide audience since. The name is the same everywhere; only the drawing of the arrowhead varies from setter to setter.
What do the buttons do?
Select a cell, then a digit from 1 to 9; the page opens on Medium. Selecting a circle or a path cell highlights the rest of that arrow as well as its row, column and box. Under the grid:
- Notes turns pencil marks on and off. A correct digit clears itself from the notes of its row, its column and its box; a wrong one leaves your notes alone, and the arrows are yours to re-read.
- Erase clears the selected cell, and pressing the same digit twice does the same.
- Undo takes back the last change to a cell, though not the notes a correct digit cleared elsewhere, and not the Mistakes count.
- Hint fills one cell with its answer, picked at random rather than the cell you could work out next.
- Solve asks first, then fills the grid and stops the timer.
- New Game deals another board at the level you are on; the level buttons deal one at theirs.
Digits 1–9, the arrow keys, Backspace or Delete and Ctrl+Z all work from a keyboard. The board is saved in this browser after every move, arrows and timer included, so it is waiting when you come back.
Where to go next
Arrow Sudoku is the sum variant that keeps its givens. Killer Sudoku takes the givens away and asks for sums in every cage instead, and Sandwich Sudoku moves the sums outside the grid, between the 1 and the 9 of each row and column. Thermo Sudoku draws lines like these but asks for an order rather than a total, so there is no arithmetic at all, and X-Sudoku is the place to go if you would rather add a region than a line.