The n=17 Square Packing Problem- Introduction

The n=17 Square Packing Problem

The image below is the most efficient n=17 square packing arrangement yet discovered. It was proposed by John Bidwell in 1998. It is unbeaten, but not proven optimal.

As a valid square packing arrangement, it contains squares which do not overlap, inside a larger square container. The container is the width/height of ~4.675530 squares.

It won’t have escaped the reader’s attention that this square packing arrangement is quite messy. It has 10 axis-aligned squares, 6 squares tilted at a ~39.8° angle, and a single square at a ~53.4° angle.

An interesting feature of the n=17 square packing problem is that the best provable lower bound on the size of the container is ~4.4452 squares. That means that the highest lower bound discovered so far is very nearly a quarter of a square’s width smaller than the best known solution.

The best known lower bound more precisely:

I thought it would be fun to spend some time looking into which offers so much room for improvement: the smallest known arrangement, or the greatest known lower bound?