NRICH: Hollow Squares

Which armies can be arranged in hollow square fighting formations?

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:
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?

Age 14 to 16

Algebra & Pre-Algebra, Quadratic Equations
High School, Educator

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