Arrow Sudoku: rules, tips and free puzzles online

In arrow sudoku, each circled cell is the sum of the digits along its arrow. Play arrow sudoku online free here, with four ranks and hints from Professor Hoot.

Arrows reward a bit of bounding: how small can the sum be, how large can the circle be? Squeeze those limits and digits fall out.

The rules

• Normal sudoku rules apply: 1 to 9 once in every row, column and 3×3 box.
• The digit in an arrow’s circle equals the sum of the digits along its arrow.
• Digits may repeat on an arrow if sudoku rules allow, that is, if the repeated cells don’t share a row, column or box.
• Arrows can bend, including diagonally. Here each circle is a single cell and each arrow is two to four cells long.

How to start

Bound the circle

The circle is at least the smallest possible sum along its arrow. If the arrow’s cells all share a house, they are all different, so two cells sum to at least 1 + 2 = 3 and three cells to at least 1 + 2 + 3 = 6. A four-cell arrow can’t be all different at all (1 + 2 + 3 + 4 = 10 is too big for a circle), so it must contain a repeated digit.

Bound the arrow

Work the other way too. The circle is at most 9, so on a three-cell arrow in one house each cell is at most 9 − (1 + 2) = 6. Big digits rarely live on arrows; small ones rarely live in circles.

Watch circles that share a house

When two circles share a row or box they must differ, and their arrows must therefore have different totals. That is often the nudge that settles an ambiguous pair.

Common mistakes

• Assuming arrow digits must all differ. They only differ when they share a house.
• Counting the circle as part of its own sum. The circle is the total, not a term.
• Forgetting that the circle and its arrow cells may share a row or box, in which case they must differ too.

How the ranks scale difficulty

Every puzzle type on Hoot’s Puzzles comes in four ranks, named after the stages of a mathematical idea. Lemma is the warm-up: about a dozen more digits are given than logic strictly needs, and singles plus the variant’s own rule will carry you through. Proposition needs a real idea or two, typically pointing and claiming. Theorem is near-minimal, so almost every clue is load-bearing and you may need pairs. Conjecture gives the fewest digits and sparser rule clues: expect to think hard.

Higher ranks use fewer arrows, around ten at Lemma and around six at Conjecture, so each remaining arrow carries more of the logic.

There are 99,999 puzzles per rank, each with its own number, so you can replay one or share it. Every type also has a 30-puzzle collection that climbs from Lemma to Conjecture. Every puzzle is checked to have exactly one solution that you can reach by logic alone, with no guessing.

Keep exploring

• Thermo SudokuDigits strictly increase from bulb to tip. Count the cells, bound the ends.
• Killer SudokuCages with sums instead of given digits. Partition arithmetic and the rule of 45.
• Little Killer SudokuOutside clues add up the diagonals their arrows point along.
• Solving techniquesSingles, pairs, cage arithmetic, line logic and Latin squares, all in one guide.
• All puzzle guidesEvery puzzle type on Hoot’s Puzzles, with rules and tips.