# Polyking Integration

## Introduction

A *polyking* is a set of squares in the square grid,
joined edge to edge or corner to corner.
It differs from a polyomino,
whose cells must be joined edge to edge.
For example,
there is only one monoking, the monomino.
There are two dikings:

To *integrate* a polyking is to arrange copies of it without
overlapping to form a polyomino.
Here I show minimal known integrations of polykings.
The solutions shown are not necessarily uniquely minimal.

Not every polyking can be integrated.
The smallest that cannot are octakings.
Here are some examples:

Where several solutions for a polyking have the same number of tiles,
I prefer one whose tiles do not cross.

### Crossless variant

#### Impossible

At most 2 tiles are needed to integrate any pentaking.

### Crossless variants

#### Impossible

There are 524 hexakings, too many to show here.
Instead I show only hexakings of which 3 or more copies are needed.
### 3 Tiles

### 4 Tiles

### 6 Tiles

### Crossless variants

Last revised 2022-05-08.

Back to Polyform Exclusion,
Equalization, Variegation, and Integration
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Polyform Curiosities

Col. George Sicherman
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