Arrow Sudoku: Rules and Solving Strategy

Learn the circle and arrow sum rule

Arrow Sudoku follows the normal rule that every row, column, and 3 by 3 box contains 1 through 9 exactly once. Each arrow begins at a circled cell, and the digit in that circle equals the sum of the digits in the cells along the arrow shaft. The circled digit is not added to the shaft; it is the total. If a circle contains 8 and the arrow has three cells, those three cells must add to 8. Digits on an arrow may repeat only when ordinary Sudoku rules allow them to occupy separate non-peer cells. Many arrows remain inside one box or row, where repeats are already forbidden. Read the drawing carefully when several arrows pass nearby. A shaft belongs only to its own circle unless the puzzle explicitly connects them. Shapedoku uses unordered shapes rather than numerical values, so its classic boards do not use arrow sums, but its scanning skills remain useful preparation.

Use arrow length to limit the circle

The number of shaft cells creates immediate bounds. A two-cell arrow has a minimum sum of 3 when its digits must be distinct, using 1 and 2. Its circle therefore cannot be 1 or 2. A three-cell arrow inside one unit has a minimum of 6 from 1, 2, and 3. Since a standard circle contains only one digit from 1 to 9, long arrows require very small shaft digits. A four-cell arrow constrained to distinct digits can total at least 10, which is too large for a single circle, so such a layout must include cells that do not all see one another or use a different arrow convention. Always combine the geometry with the stated rules rather than applying a memorized minimum blindly. Once the circle is known, list only combinations that reach that total. For a circle of 7 with two distinct shaft cells, possibilities include 1 and 6, 2 and 5, or 3 and 4.

Let crossing units choose between combinations

A combination is only the beginning because the digits still need positions. Check each shaft cell against its row, column, and box. If a 7-arrow needs either 2 plus 5 or 3 plus 4, a crossing column that already contains 2 can remove the first combination from one position and may eliminate it entirely. Parity can help as a quick filter: an odd circle total requires an odd number of odd contributions, while an even total requires an even number of odd contributions. Use parity to organize possibilities, not to replace exact checking. When arrows overlap a crowded box, inspect the circle and shaft as a small arithmetic unit, then return to classic hidden singles and pairs. One resolved arrow digit often changes several ordinary candidates. Keep Notes complete on difficult boards because a missing candidate can make a valid sum combination disappear from your analysis.

1. Identify each circle and every cell on its arrow shaft.
2. Write the minimum and maximum possible shaft sum.
3. List combinations that equal the circle or test possible circle totals.
4. Remove combinations that violate row, column, or box rules.
5. Place a digit only when every surviving combination agrees on it.
6. Rescan the surrounding units after each resolved arrow cell.