Constructed Regions

Divide the board into connected regions of the stated size. Each region contains different digits. Borders separate regions; joins put cells in the same region.

Parameters

Parameter	Meaning	Default
size	The number of cells in each constructed region or cage.	Required.
borders	Pairs of neighbouring cells that must belong to different regions.	Required.
joins	Pairs of cells that must belong to the same region.	Required.

Maths

Class: existential.

The shared predicates define the notation and every named predicate.

∃P:set<set<cell>>: partition(P,C)
  ∧ (∀R∈P: |R|=size ∧ connected(R) ∧ distinct_D(R))
  ∧ (∀p∈joins: |p|=2 ∧ samePartition(P,p[0],p[1]))
  ∧ (∀p∈borders: |p|=2 ∧ ¬samePartition(P,p[0],p[1]))

There is a partition of the board into orthogonally connected sets of size cells. Each set has distinct digits; each border separates sets and each join identifies one set. With size equal to the alphabet size, each set contains every symbol.

Example

---
format: open-rules
title: Constructed Regions example
setter: Example
source: ""
case: ""
library: https://rules.sudoku-sleuth.com/
status: read
board:
  rows: 1
  cols: 3
  symbols:
    - 1
    - 2
    - 3
---

```rules
constructed-regions size 3 borders [] joins []
```

Satisfies the rule: r1c1=1, r1c2=2, r1c3=3. The three connected cells form one region with different digits.

Breaks the rule: r1c1=1, r1c2=1, r1c3=3. The only possible three-cell region repeats 1.