How to Solve Diagonal (X) Sudoku on Paper with a Pen

Diagonal Sudoku — the X-Sudoku variant — keeps the classic 9x9 rules and silently adds a third and fourth house across the board: nine cells running corner to corner, each of which must contain every digit 1–9 exactly once. On a screen those diagonals are vaguely shaded and easy to ignore. On paper they are the entire strategic change, because the constraint does not sit in tidy 3x3 chunks — it cuts diagonally through every box the puzzle touches.

For the pen solver this is the first variant in the series where drawing on the grid is a legal move. You will mark the diagonals in red before a single digit is placed, per the method in the pen-and-paper Series Hub, and that one line changes how your eye routes every subsequent scan.

The diagonal rule on paper: the two 3x9 diagonals — R1C1..R9C9 and R1C9..R9C1 — are houses like any other. Trace them in red and audit them in the final check like a row or a column.

1. The Rules, Stated Precisely for Paper

• Place digits 1–9 so every row, column, and 3x3 box contains each digit exactly once.
• The main diagonal from R1C1 to R9C9 must contain digits 1–9 exactly once.
• The anti diagonal from R1C9 to R9C1 must contain digits 1–9 exactly once.
• A valid puzzle has exactly one solution, and the centre cell R5C5 belongs to both diagonals — a placement there eliminates from two extra houses at once.

Count the houses carefully: nine rows, nine columns, nine boxes, and two diagonals. Eleven houses audit in the final check, not nine — and because the diagonals cut across boxes, a digit placed on a diagonal is often visible in the endgame when every row and column looks "settled".

2. Paper Adaptations: Mark the Diagonals Before Solving

Before placing any digit, pick up the red pen and trace both diagonals through the grid. In print the diagonals are often faint shading that smudges under contact; a definitive red line survives the whole solve and, crucially, teaches your eye to treat the diagonal as a scan target. Trace corner-to-corner through the centre of each cell, not along the cell borders — the line must read as "these nine cells are one house".

3. The Diagonal Workflow: Three Passes

4. Diagonal-Specific Techniques

4.1 The Diagonal Single Scan

Each diagonal is a nine-cell house sliced across the puzzle, so it picks up digits from nine different boxes. When you place a digit anywhere, check whether it is the first occurrence of that digit on a diagonal. If it is, the diagonal's eight other cells can no longer hold it — and the diagonal, having only nine cells, frequently leaves one last home for a badly needed digit. This is simply "house of nine", applied along a line your pencil has already drawn.

4.2 Diagonal Hidden Singles

A hidden single on a diagonal is a digit that fits into only one cell of the diagonal, even though that cell still holds several row or column candidates. Because the diagonal crosses every line it meets, a candidate is often blocked from a diagonal cell not by its row or column — but by the diagonal itself. Scanning the red line for "eight of nine digits accounted for, ninth has one home" produces placements classic scanning never sees.

4.3 Candidate Stripping via Diagonal Placement

The momentum move of X-Sudoku: commit a digit on either diagonal, then strike that digit from every other open cell on the same diagonal, wherever those cells live. The central R5C5 cell is the jackpot — a placement there is simultaneously a placement in a row, a column, a box, and both diagonals, removing the digit from four intersecting houses. On paper, circle these centre placements in your second pen and let their downstream strikes cascade.

5. Reading the Diagonal Board

The grid below shows givens in bold, the two diagonals tinted, and two live diagonal-peer eliminations in red: R2C2 loses candidate 4 because R1C1 holds a 4 on the main diagonal, and R3C7 loses candidate 3 because R2C8 holds a 3 on the anti diagonal. The centre R5C5 shows a committed placement that strips candidates from row, column, box, and both diagonals at once.

Read the pair of eliminations as a warning: R2C2 would happily hold a 4 by row, column, and box logic, and R3C7 would hold a 3 the same way — but both cells sit on a red line that already owns that digit. Without the traced diagonals, these two placements are classic false positives; with them, they are instant strikes.

6. The Diagonal Mistakes That Kill Boards

• Trusting the 45-sum audit without the diagonals. Nine rows, columns, and boxes pass their 45 checks while a duplicated digit hides on one red line. Audit the diagonals as their own houses.
• Forgetting to strike diagonal peers. A placement's diagonal consequences are its most distant cells. Place a digit, then sweep the full red line before writing any candidate elsewhere.
• Skipping the trace on duplicates. Copying a puzzle that already repeats a digit on a diagonal makes it unsolvable before you start — check both red lines in the transcript clear step.
• Treating R5C5 as a single house. The centre cell's placement fires into four houses. Write it in pen, circle it in the second colour, and let its four-way strike ripple — this is the board's highest-leverage cell.

7. Practise the Diagonal Scan on Plain Grids First

Because diagonal logic is additive, sharpen the base eye-scan on Easy and Medium grids, then impose the "trace a red house" habit on Hard and Expert boards. When a board fights back, the diagonal is usually the delivery system for the decisive hidden single — and the fish logic that unlocks diagonal endgames is the same X-Wing structure used on classic grids. Full technique coverage lives in the strategy index.

Once the diagonal reads as naturally as a row, the next variant replaces the two long houses with nine weird ones. See Jigsaw (Irregular) Sudoku on paper to learn how to scan shapes instead of boxes.