Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### Advanced mathematics

### For younger learners

# Hollow Squares

**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:

## You may also like

### DOTS Division

### Sixational

Or search by topic

Age 14 to 16

Challenge Level

*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.

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?*

*With thanks to Don Steward, whose ideas formed the basis of this problem.*

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

The nth term of a sequence is given by the formula n^3 + 11n. Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.