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What goes in the circle?
Level 3 · Lines
How to solve arrow sudoku
The digits along an arrow add up to the digit in its circle.
An arrow starts at a circle and runs across one or more cells. The digits on the arrow add up to the digit in the circle. Normal sudoku rules still apply.
Because the circle holds a single digit, an arrow can't total more than 9. That cap keeps arrow digits small and pushes circles toward the high digits.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Digits along an arrow sum to the digit in its circle.
- Arrow sums.
- Digits may repeat on an arrow if allowed by other rules.
Worth memorizing
- The circle is at least the smallest sum the arrow can make. Two cells in one box make at least 1 + 2 = 3.
- Long arrows force small digits: four cells in a box sum to at least 10, too big for one circle, so they must cross boxes.
- Some puzzles use two-cell “pill” circles, which read as a two-digit number.
Solving tips
Find the smallest sum
Two arrow cells in the same box must be different digits, so they add to at least 1 + 2 = 3. Three cells in one box add to at least 6, so the circle is 6 to 9.
Long arrows leave the box
Four different digits add to at least 1 + 2 + 3 + 4 = 10, too big for a circle. A four-cell arrow must repeat a digit, so its cells can't all share one box.
Small circles have few options
A two-cell arrow in one box with a circle of 3 must be 1 and 2. A circle of 4 means 1 and 3, since 2 + 2 would repeat.
Read pill circles as numbers
Some puzzles use a two-cell oval called a pill. It reads as a two-digit number, so a pill showing 1 and 5 means the arrow adds to 15.
A worked example: two arrows in one box
One 3x3 box has two circles, and every arrow cell stays inside that box. The first circle is 7 with a three-cell arrow. The second circle has a two-cell arrow, and the second circle's row already has a 9 outside the box.
- The first arrow holds three different digits that add to 7. The only set is 1 + 2 + 4.
- The second arrow needs two more digits from what the box has left: 3, 5, 6, 8 and 9. The smallest pairs are 3 + 5 = 8 and 3 + 6 = 9. Every other pair is 11 or more, too big for a circle. So the second arrow holds a 3, and its circle is 8 or 9.
- The second circle's row already has a 9, so the circle is 8, and its arrow is 3 and 5.
- The box has now used 1, 2, 3, 4, 5, 7 and 8. The last two cells hold 6 and 9.
- Notice the order of the work. The three-cell arrow went first because its total allowed only one set. Its digits then took away most of the small digits the second arrow could use, and one digit in a row settled the rest.
Practice
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These cells are all in one box. Tap every digit that could go in the circle.
Select all that apply
Reference table
| Circle | Two arrow cells | Three arrow cells |
|---|---|---|
| 3 | 1 + 2 | none |
| 4 | 1 + 3 | none |
| 5 | 1 + 4, 2 + 3 | none |
| 6 | 1 + 5, 2 + 4 | 1 + 2 + 3 |
| 7 | 1 + 6, 2 + 5, 3 + 4 | 1 + 2 + 4 |
| 8 | 1 + 7, 2 + 6, 3 + 5 | 1 + 2 + 5, 1 + 3 + 4 |
| 9 | 1 + 8, 2 + 7, 3 + 6, 4 + 5 | 1 + 2 + 6, 1 + 3 + 5, 2 + 3 + 4 |
Going deeper
Arrows that leave the box
Once an arrow crosses a box border, digits can repeat. Two arrow cells may match when they share no row, column or box. A two-cell arrow bending diagonally into the next box could be 2 + 2 = 4.
To find a circle's floor, group the arrow cells. Cells that share a row, column or box need different digits. Give each group its smallest digits and add them up. That sum is the lowest the circle can be.
Long arrows and pill circles
Long arrows force small digits. Four cells into a one-cell circle total 9 or less, so at most one of them can be 4 or higher, since 4 + 4 + 1 + 1 = 10.
A two-cell pill reads as a two-digit number, left to right or top to bottom. Two arrow cells add to 18 at most, so a pill of 19 or more needs at least three arrow cells. Three arrow cells in one box add to 24 at most, so a pill above 24 needs a longer arrow or cells in more than one box.
Circles that do two jobs
A circle is still a normal cell. It can sit on a line, inside a cage, or at the end of another arrow. Then its range comes from both rules, and you take the overlap.
High circles are the busy ones. A circle with a three-cell arrow in one box is at least 6, and a circle of 9 fed by two cells in one box has four options. Mark the circle's range first, then list the sets.
Arrows inside one box also feed the 45 rule. An arrow plus its circle totals twice the circle. In the example, the two arrows and their circles used seven cells worth 7 + 7 + 8 + 8 = 30, which left 15 for the last two cells, and 6 + 9 = 15 agrees.
Common mistakes
Adding the circle into the sum
The circle holds the total. Only the cells along the arrow add up to it, and the circle's own digit isn't part of the sum.
Forgetting the circle is a sudoku cell
The circle's digit can't repeat in its row, column or box. That often rules out the largest total an arrow could reach.
Reading the arrow backward
The circle sits at the tail, where the line starts. The arrowhead points away from the circle, toward the last cell of the sum.
In mixed-rule puzzles
Arrows share a lot with killer cages. A cage that holds a whole arrow and its circle totals twice the circle, so that cage total is always even. Arrows also meet thermometers and whisper lines. A cell that is both an arrow cell and a thermometer bulb gets pushed low by both rules. With little killer diagonals, an arrow cell on a clued diagonal links two sums, so subtract what you know from each. Many puzzles also say digits may repeat on an arrow if other rules allow it, which is the same as the normal sudoku rules. Read the rules text to be sure.
What the video shows
- Digits on an arrow add up to the digit in its circle.
- The circle holds 9.
- 1 + 3 + 6 = 10. Too big.
- 2 + 3 + 4 = 9. The arrow works.
Questions solvers ask
How do arrows work in sudoku?
The digits on an arrow add up to the digit in its circle. An arrow holding 2 and 5 needs a 7 in the circle.
Can digits repeat on a sudoku arrow?
Yes, if sudoku rules allow it. Two cells on an arrow can match only when they sit in different rows, columns and boxes.
What is the biggest total an arrow can have?
With a one-cell circle, the arrow totals 9 at most. A two-cell pill circle can hold a two-digit total.
What is a pill in arrow sudoku?
A pill is a two-cell circle that reads as a two-digit number, such as 12 or 15.
What is the smallest digit that can go in an arrow circle?
With two arrow cells that must differ, the circle is at least 3. With three arrow cells in one box, it's at least 6. If two arrow cells can repeat, the floor for two cells drops to 2 (1 + 1).
Can two arrows share one circle?
Yes, in most puzzles that draw it. Each arrow adds up to the circle on its own. A circle of 9 with two separate two-cell arrows needs two pairs that each make 9. Check the puzzle's rules to confirm.
Does a pill circle read left to right?
Yes. A horizontal pill reads left to right, and a vertical pill reads top to bottom. A pill showing 1 then 4 means the arrow adds to 14.
Record Highs
Treasure Map
Six Rules Four Clues
Stretching
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