Disjoint Groups Sudoku: Rules and Solving Strategy

See the nine hidden disjoint groups

Disjoint Groups Sudoku keeps every classic rule: each row, column, and 3x3 box contains the digits 1 through 9 exactly once. It adds nine more units based on relative position inside the boxes. Take the top-left cell of every 3x3 box. Those nine cells form one disjoint group and must also contain 1 through 9 once. The nine top-middle cells form another group, as do the top-right, middle-left, center, middle-right, bottom-left, bottom-middle, and bottom-right positions. Each cell therefore belongs to four units: one row, one column, one box, and one disjoint group. The groups are spread across the board rather than drawn as connected shapes, which is why beginners often forget them after the opening scan. Shapedoku itself uses only the standard three unit types. If its glowing shapes were used in a disjoint variant, each relative-position group would need all nine shapes once just like a row or box.

Build a reliable visual map

Before solving, choose a way to identify relative positions consistently. You can number the cells inside every box from one to nine in reading order, or use names such as top-left and center-right. Then scan one disjoint group at a time. The center group is usually easiest to see because it contains the central cell of each box. For every placed digit, remove that digit from the same relative position in the other eight boxes. This elimination reaches cells that may share no row, column, or box with the original placement. On paper, a light dot or tiny position number can help prevent mistakes. On screen, mentally highlight the same square within each box before moving to another group. Do not confuse a disjoint group with a band or stack. A group selects one cell from every box, not three full rows or columns.

Use disjoint singles and subsets

Treat each disjoint group exactly like another Sudoku unit. If eight digits are already present across the nine matching positions, the last cell takes the missing digit. A hidden single appears when one digit has only one legal position in the group, even if that cell has several candidates from the perspective of its row or box. Pairs and triples also work. If two cells in one disjoint group can contain only 3 and 8, those digits can be removed from the other seven cells in that group. The extra unit can also strengthen classic interactions. Suppose a box has two candidates for 5, but one is excluded because another box already places 5 in the same relative position. The remaining box candidate becomes solved. Conversely, a disjoint pair may simplify crossing rows enough to create a normal hidden single. Full Notes are useful because candidates must be checked against four units rather than three. A placement in Disjoint Groups Sudoku affects more cells than a classic placement, so updates should follow a fixed order. Remove the digit from its row, then its column, then its box, and finally its disjoint group. After cleanup, inspect all four affected units for singles and subsets. This routine prevents the extra constraint from becoming an occasional afterthought. If you become stuck, scan by relative position rather than by digit: check all top-left cells, then all top-middle cells, and continue through the nine groups. A different scan order often reveals a hidden single that ordinary row-by-row checking misses. Once a disjoint deduction produces a placement, return to classic Sudoku techniques before hunting another specialized pattern. The variant feels dense at first, but the added rule is simply nine more all-different units. The challenge is remembering where they are, because apparently visible boxes were considered far too generous.

1. Label the nine relative positions inside every 3x3 box.
2. Treat matching positions across the boxes as complete units containing 1 through 9.
3. Remove each placed digit from the same relative position in all other boxes.
4. Search every disjoint group for hidden singles, pairs, and triples.
5. Update row, column, box, and disjoint group after each placement.
6. Switch to a position-by-position scan whenever normal row scanning stalls.