hollow square

in Napoleonic and other battles a hollow square was a popular formation for an infantry battalion e.g. Wellington's army at Waterloo, to cope with cavalry charges

not Wellington's army...

a recreation of Wellington's army hollow square formations

for a battalion (normally between 700 and 1200) of 960 soldiers, how many possible hollow square arrangements are there?
state the widths for each

if you want to start off with an easier number of people, find the 3 options for each of:

• 48
• 45
• 80 

and the 4 options for 96

in order to do this you will need to find a pair of odd factors or a pair of even factors
they need to be 'compatible' to avoid halfs

e.g. for 48, you could use 6 x 8
and then use the difference of two squares, putting a - b = 6 and a + b = 8

45 is the smallest odd number that can be expressed as the difference of two squares in three different ways (not allowing zero)
and 48 is the smallest even number

960 soldiers in a hollow square formation have many options,
10 (I'm fairly sure)
e.g. 24 x 40