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 below 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.
Age 14 to 16
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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 below 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.
Age 14 to 16