Hollow Squares printable worksheet

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. 

 

Alison's Method:



Charlie's Method:

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.