Hidden Modular

Choose the span by its stated extent. Every three consecutive digits in that span have different remainders when divided by 3.

Parameters

Maths

Class: with aggregates.

The shared predicates define the notation and every named predicate.

|𝔄|>0 ∧ ∃a,b∈[0,|cells|):
  a<=b ∧ ((extent="to-max" ∧ a=0 ∧ firstOccurrence(cells,𝔄[|𝔄|-1],b))
    ∨ (extent="between-ends" ∧ ∃i,j∈[0,|cells|): firstOccurrence(cells,𝔄[0],i)
       ∧ firstOccurrence(cells,𝔄[|𝔄|-1],j) ∧ a=minimumPosition(i,j) ∧ b=maximumPosition(i,j))
    ∨ (extent="first-x" ∧ a=0 ∧ |cells|>0 ∧ 1<=D(cells[0])<=|cells| ∧ b=D(cells[0])-1))
  ∧ ∀s∈[a,b-1): distinctAt(window(cells,3,s,false),λc:cell: classMod3(D(c)))

Choose the span from the clue through the maximum (to-max), between minimum and maximum inclusive (between-ends), or the first D(cells[0]) positions (first-x). Every triple in that span has three distinct remainders modulo 3.

Example

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

```rules
sudoku 9x9
hidden-modular r1c1-r1c2-r1c3 extent "first-x"
```

Satisfies the rule: r1c1=3, r1c2=1, r1c3=2. The first digit selects a three-cell span with all three remainders.

Breaks the rule: r1c1=3, r1c2=1, r1c3=4. The selected span repeats a remainder.

A missing endpoint or an invalid first-x extent cannot select a span, so the rule is false.