Copyright © University of Cambridge. All rights reserved.

## 'Hollow Squares' printed from http://nrich.maths.org/

In Napoleonic battles a hollow square was a popular formation for an infantry battalion designed to cope with Cavalry charges.

For example, the picture on the right shows a recreation of Wellington's army at Waterloo.

Below are two diagrams showing symmetrical hollow square formations.

**How could you quickly work out the number of dots in each?**

Click below to see two methods of dividing up the dots that might help you work it out:

**A general has**** 960 soldiers. ****How many different ways can he arrange his**** battalion**** in a symmetric hollow square?**
What can you say about battalion sizes that

**can't** be arranged as symmetric hollow squares?

**Can you find a general strategy for arranging any possible battalion into all possible symmetric hollow squares?**
What about hollow squares that are not symmetric...?

*You may also like to take a look at What's Possible?*
*With thanks to Don Steward, whose ideas formed the basis of this problem.*