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?
A general has 960 soldiers. How many different ways can he arrange his battalion in a symmetric hollow square?
Click below to see two methods of dividing up the dots that might help you work it out:
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.