Houses and regions

This page covers the rules that say where digits may not repeat: houses (6.1) and regions (6.2).

6.1 Houses: grid, box_grid, box, row, column

{"type": "box_grid", "origin": "r1c1", "box": [3, 3]}
{"type": "grid", "origin": "r1c1"}
{"type": "box", "cells": ["r1c1", "r1c2", "r1c3", "r2c1", "r2c2", "r2c3", "r3c1", "r3c2", "r4c1"]}
{"type": "row", "cells": ["r1c1", "r1c2", "…"]}

A house holds no repeated digit; a house of exactly D cells holds every digit once. Houses come only from these rules: nothing is implied, so a board without house rules has no houses.

• grid: a sudoku grid, the D × D area whose top-left cell is origin. It gives the area's D rows (top to bottom) and D columns (left to right). Every cell of the area MUST exist.
• box_grid: a grid whose boxes are a regular tiling. box = [box height, box width], with box height × box width = D (a 6×6 is box [2, 3] or [3, 2]). It gives the grid's rows, columns, then its D boxes (reading order of their top-left cells). A classic is one box_grid; a samurai is five; a super samurai is one per grid.
• box: one box with explicit cells, any shape. A jigsaw is a grid plus one box per piece.
• row / column: explicit houses for rows and columns that are not those of a D × D area (staircase, toroidal, sujiken).
• A house with the same cell set as an earlier house is the same house (overlapping grids share boxes and lines).
• Boxes, from any of these rules, are what a renderer draws with thick lines. The grid / box_grid areas also limit negative constraints and battenburg windows to one grid (6.14).

Board	Houses
classic, 6×6, 16×16	one box_grid
jigsaw	one grid + D box rules
Latin square	one grid
samurai, twodoku, butterfly, …	one box_grid per grid
samurai with a jigsaw corner	four box_grid + one grid + D box rules
toroidal, staircase	row / column / box rules

Canonical writers emit a box_grid for every D × D area whose rows, columns and regular boxes are houses, a grid where the area's boxes are irregular or absent, and box / row / column rules for the remaining houses.

6.2 region

{"type": "region", "kind": "diagonal", "cells": ["r1c1", "r2c2", "r3c3", "r4c4", "r5c5", "r6c6", "r7c7", "r8c8", "r9c9"]}

• No digit repeats among the cells. Fewer than D cells is allowed; "contains every digit" follows only when the region has exactly D cells.
• kind: diagonal | window | disjoint | colour | extra | region. To a solver all kinds mean the same; the kind is a drawing and grouping hint: diagonal (a straight diagonal: X-sudoku, argyle lines; cells listed from the top), window (windoku windows), disjoint (same position in every box), colour (colour groups), extra (shaded extra regions: asterisk, centre dot, …), region (anything else).