ç Mayhematics

# Chessboard Dissection Problems

## Contents

### Introduction

The Square Pieces: Polyominoes — Historical Introduction — Aesthetic considerations

Dissection Problems in PFCS/FCR — Summary of results in chronological order

The FCR Coding Method — Key to decoding the dissection problems in Fairy Chess Review

Enumeration of Square Pieces — The binary number method

### Smaller Pieces

The 1-Square Piece: Monomino — The problems of 'Squaring the Square' and 'Squaring the Rectangle'

The 2-Square Piece: Domino — Dominizing the Chessboard — a study of dissections into 2-square pieces ('dominoes')

### The 5-Square Pieces: Pentominoes

The 5-Square Pieces: Pentominoes — Shapes formed with the 12 5-square pieces.

[5]s + [4]s — Shapes formed with the 17 pieces of 5 and 4 squares, and subsets.

[5]s + [4]s + [3]s + [2] + [1} — Shapes formed with the 21 pieces of 1 to 5 squares, and subsets.

[5]s on the Chessboard — Where the four uncovered squares form various disconnected patterns.

[5]s + [4] on the Chessboard: transfers — Where the uncovered squares are connected. — Proof of Dawson's theorem.

[5]s + [4] on the Chessboard: patterns — Inset squares, rectangles, triangles, etc.

### The 6-Square Pieces: Hexominoes

The 6-Square Pieces — Shapes formed with the 35 hexominoes.

The [6]s in Multiple Shapes — Two or more shapes formed simultaneously with the 35 hexominoes.

Using more than 35 [6]s — Includes Frans Hansson's 6-fold magnified [6]-shapes using one duplicate.

Using less than 35 [6]s — Includes shapes using the 20 asymmetric pieces, the 24 evenly chequered pieces, etc.

The [6]s + [5]s — Shapes formed with all 47 pieces of 5 or 6 squares.

### The Larger Pieces

The 7-Square Pieces: Heptominoes — Shapes formed with the 108 heptominoes. [This page activated 23 April 2014]