Sudoku X strategy: let the diagonals do real work

The X adds houses; it does not replace boxes

Sudoku X is ordinary Sudoku with two additional all-different houses: the main corner-to-corner diagonals. Bill Murphy’s published diagonal puzzle explicitly retains the usual rules and adds the marked diagonals. [1] A completed diagonal therefore contains each digit from 1 to 9 once.

The word “main” matters. Short sloping runs parallel to those diagonals are not new houses. Two cells that look diagonally adjacent may have no special relationship at all. Follow the marked lines, not a general idea that Sudoku X forbids every diagonal repetition.

Andrew Stuart’s dedicated Sudoku X solver treats the diagonals as extra constraints. [2] Its existence also highlights a practical point: a classic-only solver or candidate list can miss legal deductions for this variant. Read the selected rules before using any tool as a check.

Know which cells acquire extra neighbours

Using row-column notation, the descending diagonal is r1c1, r2c2 and so on to r9c9. The other is r1c9, r2c8 through r9c1. Their union has 17 cells because r5c5 appears on both lists. This count is a direct consequence of the two nine-cell lines sharing one centre.

A cell outside the X still checks its usual three houses. A noncentral diagonal cell checks four, and the centre checks five. These are memberships, not a guarantee that the centre solves first. The actual givens determine which constraint carries useful information.

The centre is constrained, not preassigned

Suppose ordinary row, column and box checks leave r5c5 with {2,5,8}. A confirmed 2 elsewhere on one main diagonal removes 2. A confirmed 8 on the other removes 8. The centre is now 5. Each removal comes from an explicit peer relationship.

Nothing in the rules says the centre must always be 5. Digits are not arranged in numerical order along a diagonal. A diagonal can contain 9 followed by 1, and a finished grid need not look symmetric. The shaded X describes membership, not a numerical pattern or an instruction to solve outward from the centre.

The original classic constraints remain mandatory, as in Nikoli’s rule definition. [3] A diagonal placement that duplicates a row digit is still wrong, even if the diagonal itself looks perfect.

Move from a diagonal into a box

Assume the only surviving positions for 6 on the main diagonal are r1c1 and r2c2. One of them must be 6, but the diagonal does not yet choose which. Both lie in the upper-left box. That box therefore cannot put 6 into any of its seven other cells.

Check every diagonal cell before making that elimination. If r8c8 also allowed 6, the digit could escape the box and the argument would fail. This is the same logic as a familiar box-line interaction, applied to an added house.

The reverse can work too: when every possible 6 in a box lies on one main diagonal, eliminate 6 from that diagonal outside the box. The common intersection is what matters, not the angle of its cells.

Give each new placement a second pass

After placing a digit on the X, scan the entire affected diagonal for candidate removals. Then scan its ordinary houses. A deletion near the far corner may create a single, whose row can affect cells nowhere near the original move. This is how a local diagonal clue enters the rest of the puzzle.

Avoid confusing Sudoku X with X-Wing. Sudoku X changes the puzzle’s rules. An X-Wing is a candidate pattern that can appear in ordinary Sudoku; its name does not authorise diagonal constraints. Before applying either idea, state whether you are using a printed rule or a deduction from candidates.

A useful error check is to write two short lists: the digits already on each main diagonal, and the digits still missing from it. If a missing digit has no candidate anywhere on its diagonal, revisit the last candidate deletion or placement. The contradiction does not mean the diagonal is optional. Conversely, several surviving candidates simply mean that this house has not yet provided a single; continue with another justified deduction.

Test the boundaries of the rule

The ZUDOKU Sudoku X page currently presents a Plus preview. The following rule check is interactive here; it does not require that preview to be playable. For a full constructor example, use the cited diagonal puzzle and its own instructions.