---
name: Doll Sum
url: https://rules.sudoku-sleuth.com/doll-sum.md
kind: doll-sum
parameters:
  cells:
    description: The marked cells, for example r1c1-r1c2-r2c2.
    default: Required.
    type: list<cell>
  sums:
    description: The permitted or required totals.
    default: Required.
    type: list<number>
---

# Doll Sum

For each listed total, two digits sum to that total and all digits strictly between them also sum to it.

## Parameters

| Parameter | Meaning | Default |
| --- | --- | --- |
| cells | The marked cells, for example r1c1-r1c2-r2c2. | Required. |
| sums | The permitted or required totals. | Required. |

## Maths

Class: **with aggregates**.

The [shared predicates](https://rules.sudoku-sleuth.com/predicates.md) define the notation and every named predicate.

```text
∀s∈sums: ∃i,j∈[0,|cells|): i<j ∧ D(cells[i])+D(cells[j])=s
  ∧ sum_D(betweenPositions(cells,i,j))=s
```

For every listed sum, there are two positions whose endpoint digits sum to it. The digits strictly between those positions also sum to it.

## Example

````markdown
---
format: open-rules
title: Doll Sum example
setter: Example
source: ""
case: ""
library: https://rules.sudoku-sleuth.com/
status: read
---

```rules
sudoku 9x9
doll-sum r1c1-r1c2-r1c3 sums [5]
```
````

Satisfies the rule: r1c1=2, r1c2=5, r1c3=3. The end digits total 5, matching the digit between them.

Breaks the rule: r1c1=2, r1c2=4, r1c3=3. The digit between the ends differs from their total.