All Sudoku Variants
9×9 · Constraint

Disjoint Groups Sudoku

Also known as Disjoint Sets Sudoku

Every cell that sits in the same position within its 3×3 box forms a ninth group that must also hold 1–9.

1 5 8 3 6

An example Disjoint Groups puzzle: the shaded cells sit in the same position of every box and form one group holding 1–9 (nine such groups tile the grid).

Disjoint Groups Sudoku adds a subtle extra rule: the cells that occupy the same position within their 3×3 box form a new group that must also hold 1–9. Take the top-left cell of all nine boxes, for instance — together they must contain each digit exactly once. Since a box has nine positions, the grid gains nine of these hidden extra regions.

The rule: same seat, every box

Number the nine seats inside a 3×3 box from top-left to bottom-right. The rule says that seat number 1 across all nine boxes forms a region needing 1–9, and so does seat 2, and so on for all nine seats. Equivalently: two cells in the same position of their boxes may never share a digit. The shaded cells below are one such group — the top-left seat of every box.

The top-left cell of every box forms one disjoint group — nine cells that must hold 1–9.

Nine extra regions in disguise

It helps to picture the nine disjoint groups as nine more regions laid invisibly over the grid, each scattered rather than clustered. That scattering is what makes the rule powerful: a disjoint group reaches across all nine boxes at once, tying together cells that plain Sudoku keeps far apart. A digit placed in the top-left of one box now forbids it in the top-left of all eight others — a long-range constraint no ordinary rule provides.

How to use it

Treat each seat as its own unit. When you place a digit, immediately rule it out of the same seat in every other box. This is especially sharp when a digit is already crowded: if seat 5 (the box centre) holds a 7 in three boxes, the other six centres lose their 7 at a stroke. Scanning "by seat" alongside the usual by-row, by-column and by-box scans is the key habit for Disjoint Groups.

Solving strategy

  • Scan by seat. For each placed digit, clear the matching position in all other boxes.
  • Treat seats as regions. Hunt hidden singles within a disjoint group just as you would in a box.
  • Exploit crowded digits. A digit that already fills several boxes' matching seats is quickly forced out of the rest.
  • Combine the scans. Row, column, box and seat together often leave a digit a single legal cell.

Common mistakes

  • Forgetting the extra rule mid-solve. The disjoint groups are easy to overlook because they are not drawn — scan by seat every time.
  • Confusing seat with box. A disjoint group is one position across all boxes, not the nine cells of a single box.
  • Treating it as optional. The seat rule is a full constraint; every seat needs 1–9.
  • Dropping the ordinary rules. Rows, columns and boxes still apply on top of the disjoint groups.

Can I play it here?

Not yet — Disjoint Groups Sudoku isn't one of the puzzle types SudokuStreak generates for now. But you can learn its rules above, and there's plenty here to sink your teeth into meanwhile: