Play Arrow Sudoku Online

Arrow Sudoku: The Sum-Path Logic Puzzle

Arrow Sudoku is a compelling Sudoku variant that blends classic number-placement logic with arithmetic constraints. Arrow-shaped lines are drawn across the grid, each starting from a circle cell and extending through one or more path cells. The digit placed in the circle must equal the sum of all digits along the arrow's path. Combined with standard Sudoku rules, these arrow constraints create deeply rewarding puzzles that test both your logic and mental maths.

🤔 What Is Arrow Sudoku?

An Arrow Sudoku puzzle uses a standard 9×9 grid divided into nine 3×3 boxes, just like regular Sudoku. The key addition is a set of arrows overlaid on the grid. Each arrow consists of a circled cell (the head) and one or more connected cells forming a path (the body). The fundamental rule: the digit in the circle cell must equal the sum of the digits in all path cells belonging to that arrow.

For example, if an arrow has a circle and two path cells containing 3 and 5, the circle must contain 8. If an arrow has three path cells with 1, 2, and 3, the circle must contain 6.

📋 Rules of Arrow Sudoku

Arrow Sudoku combines standard Sudoku rules with one elegant additional constraint:

1. Standard Sudoku rules — Every row, column, and 3×3 box must contain the digits 1–9 exactly once.
2. Arrow-sum rule — For each arrow on the grid, the digit in the circle cell must equal the sum of all digits along the arrow's path. Digits may repeat on an arrow provided they don't violate standard Sudoku constraints.

Because the circle cell holds a single digit (1–9), the sum of the path cells is always between 1 and 9. This tightly constrains what can go on longer arrows. Every puzzle has exactly one solution reachable through pure logic — no guessing required.

⭐ Difficulty Levels Explained

Our Arrow Sudoku offers four difficulty levels that affect the number of given digits and arrow configuration:

• Easy — More given digits and shorter arrows (2–3 path cells). The arrow constraints quickly narrow down possibilities. Ideal for learning the variant.
• Medium — Fewer givens with arrows of 2–4 path cells. Requires combining arrow-sum logic with standard Sudoku elimination. A solid daily challenge.
• Hard — Significantly fewer clues and longer arrows (3–5 path cells). Demands advanced techniques like arrow-box interaction and cross-constraint deduction.
• Expert — Minimal given digits with many long arrows (3–5 path cells). Requires deep logical chains and expert-level Sudoku skills.

🧠 Essential Arrow Sudoku Strategies

Mastering Arrow Sudoku requires a blend of classic Sudoku techniques and arrow-specific logic:

1. Maximum Sum Analysis

The circle cell holds a single digit (1–9), so the sum of path cells can never exceed 9. On a 3-cell path, the maximum average is 3, which means most cells must be very low. On a 4-cell path, the maximum average is just 2.25 — the only possible combinations use 1s and 2s. This drastically limits candidates.

2. Minimum Circle Values

A path of n cells has a minimum sum of n (all 1s, if allowed by Sudoku rules). So a 2-cell path arrow's circle is at least 2, a 3-cell path at least 3, and so on. Conversely, the circle can't be less than the number of path cells.

3. Arrow-Box Interaction

When an arrow's path cells all lie within the same 3×3 box, no digits can repeat among them. This further constrains the possible sums and often forces specific combinations. For example, a 3-cell path in one box summing to 7 can only be {1,2,4} — since {1,3,3} and {2,2,3} have repeats.

4. Circle Cell Implications

If you determine the circle digit first (through standard Sudoku logic), you immediately know the exact sum the path cells must reach. This converts the problem into a partitioning puzzle — finding which digits add up to that sum while respecting all Sudoku constraints.

5. Standard Sudoku Techniques

Don't forget naked singles, hidden singles, pointing pairs, and other classic strategies. Arrow Sudoku is still Sudoku at its core — the arrow constraints supplement rather than replace traditional logic.

📊 Quick Reference: Arrow Path Sums

Use this table to quickly identify possible digit combinations for common arrow lengths:

• 2-cell path, circle 3: {1,2}
• 2-cell path, circle 4: {1,3}
• 2-cell path, circle 5: {1,4} or {2,3}
• 2-cell path, circle 7: {1,6}, {2,5}, or {3,4}
• 2-cell path, circle 9: {1,8}, {2,7}, {3,6}, or {4,5}
• 3-cell path, circle 6: {1,2,3} only (if no repeats)
• 3-cell path, circle 7: {1,2,4}
• 3-cell path, circle 8: {1,2,5} or {1,3,4}
• 3-cell path, circle 9: {1,2,6}, {1,3,5}, or {2,3,4}
• 4-cell path, circle 9: {1,2,3,3} — but only if repeats are allowed across houses

🆚 Arrow Sudoku vs. Other Sudoku Variants

How does Arrow Sudoku compare?

• vs. Classic Sudoku: Classic uses only given digits as clues. Arrow Sudoku adds sum-based arrow constraints, creating a richer logical puzzle.
• vs. Killer Sudoku: Killer uses cage sums where all cells contribute equally. Arrow Sudoku has a directional relationship — one cell (the circle) must equal the sum of the others.
• vs. Thermo Sudoku: Thermo requires strictly increasing digits along a path. Arrow Sudoku requires the circle to equal the path sum — ordering doesn't matter, only arithmetic.
• vs. X Sudoku: X Sudoku adds diagonal constraints. Arrow Sudoku adds path-sum constraints. They can be combined for an extra challenge!

📜 History of Arrow Sudoku

Arrow Sudoku emerged from the rich world of Sudoku variants that proliferated in the 2000s and 2010s. The concept of linking a cell's value to the sum of a connected path appeared in various puzzle magazines and competitions, drawing on similar ideas from Kakuro and Killer Sudoku.

The variant gained enormous mainstream visibility through Cracking the Cryptic, the acclaimed YouTube puzzle channel. Their engaging solving videos showcased Arrow Sudoku's elegant logic and introduced the variant to millions of viewers worldwide. Today, Arrow Sudoku is a favourite at World Puzzle Federation events and a beloved staple of the online puzzle community.

💪 Benefits of Playing Arrow Sudoku

• Sharpens mental arithmetic — constant summing and partitioning builds number fluency.
• Strengthens logical reasoning — managing arrow constraints alongside standard Sudoku rules deepens deduction abilities.
• Improves working memory — tracking sums, candidates, and relationships across the grid exercises your short-term memory.
• Visually engaging — the arrow graphics make the puzzle distinctive and satisfying to solve.
• Highly rewarding — the moment an arrow sum clicks into place and cascades through the grid is deeply satisfying.

🎮 More Sudoku Variants to Explore

• Classic 9×9 Sudoku — The original puzzle. Start here if you're new.
• Killer Sudoku — Cage sums replace given digits. Logic meets arithmetic.
• Thermo Sudoku — Digits must increase along thermometer lines.
• X Sudoku — Both main diagonals must also contain 1–9.
• Jigsaw Sudoku — Irregular regions replace the standard 3×3 boxes.