Shifter Line

Digits on the line cannot be 5. Within a region, neighbours are both odd or both even and both below or both above 5. At each region crossing, exactly one of those relationships changes.

Parameters

Maths

Class: first-order.

The shared predicates define the notation and every named predicate.

(∀c∈line:D(c)!=5)
∧ ∀i,j∈[0,|line|): consecutive(line,i,j,false) →
  let P=(odd(D(line[i]))↔odd(D(line[j]))); let Q=(high(D(line[i]))↔high(D(line[j])));
  (sameRegion(line[i],line[j]) → P ∧ Q) ∧ (¬sameRegion(line[i],line[j]) → ¬(P↔Q))

No digit is 5. Neighbours within the same authored region share parity and polarity (below/above 5). Across a region boundary exactly one of parity and polarity changes.

Example

---
format: open-rules
title: Shifter Line example
setter: Example
source: ""
case: ""
library: https://rules.sudoku-sleuth.com/
status: read
---

```rules
sudoku 9x9
shifter-line r1c1-r1c2
```

Satisfies the rule: r1c1=1, r1c2=3. 1 and 3 are both odd and below 5 within the same box.

Breaks the rule: r1c1=1, r1c2=2. 1 and 2 have different parity within the same box.