Imagine this: there are 1000 students at a school, each with a locker, which is shut. At the start of the day, the first student comes into school and opens all of the lockers.
Then the second student comes in and touches locker numbers 2, 4, 6, 8 etc... if it's shut, he opens it, if it's open he shuts it.
The third student comes in and touches numbers 3, 6, 9, 12... etc. (If it's shut she opens it, if it's open she shuts it.)
It carries on all day. The students come in, and starting with their own locker, they touch the ones which are multiples of their number.
The challenge is figuring out what state the lockers are in by the end of the day.
PHEW!
There are a couple of things to think about when tackling this problem...
1. Downsize the issue. 1000 is a big number. Try it out with 20 lockers and see if a pattern starts emerging which you can extrapolate.
2. Think about what happens to a specific locker. How many times will locker number 1 be touched? How many times will locker number 21 be touched?
It wasn't long before a few of the students in the group realised that this problem has a lot to do with square numbers - such as 1, 4, 9, 16, and so on. When you write down the factors of each locker number you will realise why.
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