How to Solve Arrow Sudoku on Paper with a Pen

Arrow Sudoku turns the classic grid into a set of tiny equations. Each arrow is a circle sitting in one cell with a straight shaft running through 2–4 neighbours, and the circled digit must equal the sum of the digits along that shaft. On screen the app can run the arithmetic for you and shade the circle as it fills. On paper, the arithmetic is yours — and that is precisely why the variant rewards ink: the moment you internalise the 3-cell minimum, half the circles on the board start collapsing by themselves.

Built on the traversal intuition you developed in the Thermometer guide, arrow logic swaps a pure ordering rule for a sum rule — but the discipline of reading the diagram in red before solving is identical. Both assume the universal tools from the pen-and-paper Series Hub: two pens, three passes, and no undos.

The two-pen rule in action: the arrow line is geometry you cannot "un-read"; once you trace it in red and circle the bulb, every future digit you place has to respect an equation, not a vague shape.

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1. The Rules, Stated Precisely for Paper

• Place digits 1–9 so every row, column, and 3x3 box contains each digit exactly once — the classic baseline never disappears.
• Each arrow has a circled cell (the circle) at one end and a straight shaft of 2–4 cells leading away from it.
• The digit in the circled cell equals the sum of the digits along every shaft cell. This is an exact arithmetic equality.
• All classic repetition rules still apply independently inside each shaft and circle — so a shaft cell repeats none of its house digits, and no digit repeats along the shaft within a single house.
• Exactly one complete solution exists; every deduction must be deterministic.

The sum rule can be read as a powerful inequality too: since no shaft cell repeats within a house, the shaft's digits are distinct small values, and the circle above them is necessarily their total.

2. The Paper Adaptations — Draw the Equations First

• Trace every arrow in red. Follow the shaft cell by cell so the exact path (and which cells the circle "sums") is unambiguous.
• Circle each bulb cell in red. That is the equation's output register — never lose sight of which cell holds the sum.
• Write "SUM" next to each circle. A tiny margin label converts the arrow from decoration into a constraint you will actually use.
• Note the shaft length. Write 2, 3, or 4 beside each arrow so the minimum-sum bound is instantly readable.

3. The Core Weapon — The 3-Cell Minimum

The single most powerful arrow fact needs no solving at all. The two smallest digits available are 1 and 2, and because no digit repeats within a house, a two-cell shaft can never contain two of the same digit. Its sum is therefore at least 1 + 2 = 3.

That one sentence strikes 1 and 2 from every circle on the board regardless of its shaft length, because a longer shaft can only raise the floor further. A 3-cell shaft sums to at least 1+2+3 = 6 (as the shaft cells are distinct within a house), and a 4-cell shaft to at least 1+2+3+4 = 10. Write the floor for each arrow's circle in the margin and strike the low digits immediately.

4. Circle = 3 and Circle = 4 — Forced Shafts

When a circle collapses to a specific small value, the shaft becomes a forced set. A circle of 3 is possible only when the shaft is exactly the two cells {1,2} — there is no other way to reach 3, and a longer shaft already overflows. Strike every candidate but those two out of the shaft and, once either resolves, the other follows.

A circle of 4 forces the shaft to {1,3}. The alternative phrasing 1+1+2 is impossible because the two 1s would repeat within the same house. The general principle: the moment a circle value is fixed, translate it into a set of exact shaft partitions and test each against house repetition.

5. Propagation — The Circle Dominates Its Shaft

A sum is always larger than any single positive addend, so every shaft cell must be strictly smaller than its circle. That yields a sweeping strike: all shaft cells lose every candidate at or above the circle's current minimum. If a circle's floor is 6, every shaft cell drops to at most 5 immediately.

This propagation is a two-way door. Once the shaft collapses to a small set, the circle's sum is bounded above (the maximum of the distinct shaft digits), shaving the circle's candidates in turn. Every arrow is a coupled system: adjust the circle, adjust the shaft; adjust the shaft, adjust the circle. Keep both directions written in red and the board steers itself.

6. Diagram — Reading the Sum Equation Off an Arrow

A circled cell (left) feeds a 3-cell shaft. The bracket labels restate the constraint: the circle must equal the total of the three shaft cells, and the close-up at each cell writes its live candidate window.

Read the windows top to bottom: because the circle's floor is 6, every shaft cell is capped at 5. If any shaft cell resolves to 5, the other two must total at least 1 with distinct digits — a fast arithmetic check on every row crossing the arrow.

7. The Three-Pass Workflow, Adapted to the Arrow

8. The Common Mistakes That Break Arrow Boards

• Allowing repeats on the shaft. A two-cell shaft summing an even circle needs two distinct digits under classic rules; the 1+1 shortcut is invalid and poisons the board.
• Forgetting the circle reads as a full house cell too. The circled digit still obeys row/column/box — it is both the sum's output and an ordinary Sudoku digit.
• Skipping the digit-dominance strike. A shaft cell holding a value equal to its circle violates "strictly smaller" and is a hidden contradiction waiting to surface.
• Ignoring the 4-cell shaft floor. A 4-cell arrow's circle can never be under 10; leaving small candidates in it is a waste of markup and invites false solves.
• Skipping the 45-check. With equations layered on houses, a subtle arithmetic slip is easy to miss. Audit every finished row, column, and box.

9. Train the Equation Instinct

Begin on Easy grids and force the SUM label on every arrow before solving. Progress through Medium, Hard, and Expert, verifying with the 45-check. The strategy index carries the classic reduction moves that resolve shaft cells once the arithmetic trims the field.

When the sum-and-dominance dance feels fluent, Anti-Knight Sudoku adds a geometric exclusion on top of the classic rules — the next step in this paper series.

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