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.
A worked example
A two-cell shaft with a circle of 3 has exactly one option: {1,2}. A circle of 4 gives {1,3}. A circle of 17 is impossible, because two digits cap out at 9+8 = 17 only when both are the largest available and they do not see each other.
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.
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.
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