# 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.

• Monoking
• Dikings
• Trikings
• Tetrakings
• Pentakings
• Hexakings

## Pentakings

At most 2 tiles are needed to integrate any pentaking.

## Hexakings

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

### Crossless variants

Last revised 2024-05-14.

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Col. George Sicherman [ HOME | MAIL ]