Arrow Sudoku: Rules & How to Solve
The circled digit equals the sum along its arrow. A clean variant with surprisingly deep deductions.
The rule
The digit in the circle equals the sum of the digits along the attached arrow. A digit may repeat along an arrow unless a row, column, or box forbids it. Normal sudoku rules apply.
Which arrows are even possible?
The circle holds a single digit, so it can never exceed 9. That caps the shaft hard. When the shaft cells all see each other they must be distinct, so the smallest total they can make is 1+2+…, and any arrow whose minimum already exceeds 9 cannot exist.
| Shaft cells | Smallest distinct sum | Smallest sum if repeats allowed | Verdict |
|---|---|---|---|
| 2 | 3 | 2 | possible |
| 3 | 6 | 3 | possible |
| 4 | 10 | 4 | impossible with a one-digit circle |
This is why long arrows in real puzzles almost always bend out of their box: once the shaft cells stop seeing each other they may repeat digits, and a four-cell shaft becomes legal again (1+1+1+1 = 4).
VariantDoku shows the same reference inside the app: open any puzzle, tap the rules sheet, and each constraint on the grid gets its own tips card.
How to solve arrow sudoku
Mind the minimum
A two-cell arrow needs a sum of at least 3 (1+2), so its circle is 3 or more. Longer arrows push the circle higher fast.
Two-digit pills
When the circle spans two cells it forms a number 10–99, letting long arrows sum to large totals — read the pill direction carefully.
Shared cells
Arrows that cross thermometers, cages, or other arrows let you substitute one constraint's value into another.
Bound the circle from the shaft
Add up the smallest digits the shaft could take and the circle cannot be below that total. A shaft already containing 6 forces a circle of 7 or more.
Watch for shafts that leave the box
If every shaft cell sees every other, the digits must be distinct. The moment a shaft bends out of its box, repeats become legal and the minimum sum drops sharply.
Step by step: techniques from the hint engine
Each example is drawn the way VariantDoku's hints show it: red candidates are eliminated, blue ones are what the step proves, and green cells are solved.
Bound the circle and the shaft
When the shaft cells all see each other they are all different, so three of them add up to at least 1 + 2 + 3 = 6, and the circle is 6, 7, 8 or 9. The same sum caps the shaft: the other two cells contribute at least 1 + 2 = 3, so no shaft cell can be more than 9 − 3 = 6.
Read a pill as a two-digit number
A two-cell circle — a pill — reads as a two-digit number. Two different shaft digits add up to at most 9 + 8 = 17, so the pill is at most 17 and its first digit must be 1. It cannot be 11, because the pill's cells share a box, so it is 12–17: the second digit is 2–7. Each shaft cell then needs a partner of at most 9 to reach 12, so it is at least 3.
A worked example
A two-cell shaft under a circle of 3 has exactly one option: {1,2}. A circle of 4 gives {1,3} — or {2,2}, but only when the two shaft cells share no row, column or box. The circle itself is a single digit, so the shaft can never total more than 9: three shaft cells that all see each other can only make 6 to 9, and a circle of 9 above them needs {1,2,6}, {1,3,5} or {2,3,4}.
Work the other direction too. If the shaft already contains a 6, the circle is at least 7, which frequently leaves only two or three candidates in the circle's own row and column.
On 4×4 and 6×6 boards
The circle stops at 4 or 6. On a 6×6 board, three shaft cells that see each other add up to at least 1 + 2 + 3 = 6, so the circle is 6 and the shaft is exactly {1,2,3}. On a 4×4 board the circle is at most 4, so a two-cell shaft is 1+2 under a 3 and 1+3 under a 4; repeats such as 2+2 are only possible when the two shaft cells share no row, column or box.
| Board | Shaft (cells all different) | Smallest sum | Circle |
|---|---|---|---|
| 4×4 | 2 cells | 3 | 3–4 |
| 4×4 | 3 cells | 6 | impossible |
| 4×4 | 4 cells | 10 | impossible |
| 6×6 | 2 cells | 3 | 3–6 |
| 6×6 | 3 cells | 6 | 6 = 1+2+3 |
| 6×6 | 4 cells | 10 | impossible |
Try one from the catalog
Real VariantDoku puzzles with this rule. Each one opens in the app.
Frequently asked questions
Can digits repeat along an arrow?
Yes, unless the cells share a row, column, or box. Only the standard sudoku rules restrict repeats along the shaft.
What does the circle mean in arrow sudoku?
It holds the total of every digit the arrow passes through.
What is a pill in arrow sudoku?
A two-cell circle that reads as a two-digit number, used so long arrows can reach sums above 9.
Can digits repeat on an arrow shaft?
Only if the cells do not see each other. Shaft cells within one row, column or box must be distinct; a shaft that bends across boxes may repeat.
Can the circle be part of the shaft?
No. The circle holds the total and the shaft holds the addends — the circled cell is never counted in its own sum.
Play Arrow Sudoku in VariantDoku
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