How to Play Arrow Sudoku
Digits on an arrow must sum to the digit in the circle
Arrow Sudoku is the variant that turned modern setters into minor celebrities: it is everywhere on Cracking the Cryptic, and for good reason. The rule fits in a breath — digits along an arrow's shaft sum to the digit in its circle — yet it generates some of the most elegant logic in all of Sudoku. The circle is a built-in ceiling: it can hold at most 9, so a three-cell shaft averages 3 at most, and long shafts are forced into tiny digits before you've placed anything. Unlike killer cages, shaft digits may repeat when rows, columns, and boxes allow it, which keeps you honest — you can never outsource the thinking to a combination chart. Arrow suits solvers who enjoyed killer's arithmetic but want it threaded through the grid geometrically, and puzzle fans who like openings that feel like proofs rather than scans. Expect easy arrows to play like medium classics and expert arrows to be genuine afternoon projects.
The Rules
Standard Sudoku rules apply: fill every row, column, and 3×3 box with the digits 1–9, each appearing exactly once.
Arrow Sudoku places arrows on the grid, each starting from a circled cell. The digits in all cells along the arrow (not including the circle) must sum to the digit placed in the circled cell. Since the maximum digit is 9, long arrows with many cells are highly constrained. Arrow Sudoku often combines beautifully with standard elimination and candidate logic.
At a Glance
How to Solve Arrow Sudoku
Worked Example: Opening a Arrow Sudoku Puzzle
Say a three-cell arrow curls inside one box, its circle at r1c3 and shaft on r2c2, r2c3, r3c2. The shaft's minimum is 1+2+3 = 6 — three different digits in one box can't do better — so the circle is 6, 7, 8, or 9. Now suppose column 3 already contains 8 and 9. The circle drops to {6,7}. Try 6: the shaft must be exactly {1,2,3} in some order. Try 7: the shaft is {1,2,4}. Either way, the shaft cells contain 1 and 2 — so 1 and 2 are barred from every other cell of that box, and any box cell that held only {1,2} as candidates is now impossible, forcing earlier decisions elsewhere. Next, check the box's remaining digits: whichever of 3 and 4 the shaft doesn't use must sit in the box's non-arrow cells, and if only one such cell can take it, the circle resolves backwards — a 4 forced outside the shaft means the shaft is {1,2,3} and the circle is 6. This is arrow solving in miniature: bound the circle from below by the shaft's minimum, from above by its houses, then let the shared digits of every surviving combination do the eliminating.
Common Mistakes in Arrow Sudoku
| Mistake | How to fix it |
|---|---|
| Assuming shaft digits can't repeat, as in killer cages | Arrow shafts may repeat digits whenever the shaft cells share no house. A two-cell shaft spanning two boxes in different rows and columns can legally be {3,3} under a 6 — prune combinations with Sudoku rules, not cage rules. |
| Ignoring the shaft's minimum sum | An n-cell shaft within one house sums to at least 1+2+…+n. Three cells means the circle is at least 6; four cells, at least 10 — impossible, unless repeats across houses lower the floor. Apply the floor before anything else. |
| Solving each arrow in isolation | The power move is intersecting arrow combinations with house logic: digits common to all surviving shaft combinations are locked into the shaft, and digits impossible in every combination are locked out. Arrows are elimination engines, not side quests. |
Common Questions
What is Arrow Sudoku? ▾
Arrow Sudoku is a variant where arrows are drawn across the grid. Every digit placed along an arrow must sum to the number written inside the circle at the arrow's base. Standard Sudoku rules still apply — every row, column, and 3×3 box must contain the digits 1–9 exactly once.
How do the arrows and circles work? ▾
Each arrow starts at a circle and extends through one or more cells. The digit in the circle is the target sum for all cells the arrow passes through. A single-cell arrow tells you the exact digit in that cell. Longer arrows have multiple possible combinations, which you narrow down using Sudoku constraints.
Is Arrow Sudoku harder than regular Sudoku? ▾
It depends on your experience. Arrow Sudoku adds an arithmetic layer on top of standard logic. Easy puzzles are comparable in difficulty to medium classic Sudoku. Hard and expert arrow puzzles require simultaneous constraint-tracking that most players find significantly more challenging than any classic Sudoku difficulty.
Can a digit repeat along an arrow? ▾
No — digits can repeat on an arrow only if standard Sudoku rules allow it. Because each row, column, and box must contain 1–9 once, a digit can appear twice on an arrow only if the two cells are in different rows, columns, and boxes. In practice, most arrow cells are highly constrained.
What is the best first move in Arrow Sudoku? ▾
Start with single-cell arrows — the digit in that cell equals the circle value directly. Then look for short arrows with extreme sums. A 2-cell arrow summing to 3 can only be {1,2}; summing to 17 can only be {8,9}. These give you the tightest starting constraints without any guessing.
How long does Arrow Sudoku take to solve? ▾
Easy puzzles take 5–15 minutes. Medium puzzles typically run 15–30 minutes. Hard puzzles average 30–60 minutes, and expert puzzles can take over an hour even for experienced solvers. Solve times improve quickly once you develop intuition for which arrow combinations are possible at a glance.