Maths Investigation 3

The Locker Problem
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.

Maths Investigation 2

Today we looked at a bit of Geometry.
Firstly Polygons: How many diagonal lines are there inside polyhedral shapes? And is there a general rule? Can the number be predicted?
We found that for a four-sided shape the total is 2, five-sided it's 5, six-sided it's 9, seven-sided it's 14, and eight-sided it's 20. So obviously the number of diagonals increases. To find a rule, we looked carefully at each corner, and we found it useful to notice, for every shape, how many diagonals came from each corner.
In the end we did come up with a rule, and used it to predict that the number of diagonals in a 100-sided shape is 4850 (phew!).
Here's our rule, where "s" means the number of sides on the shape:
(s-3) x s / 2. We checked it and it works for all the smaller shapes, so we think it will work for the larger ones too.

Then we had some time to work with Polyhedra. We looked at three nets and noticed that although they are all called pyramids, they were quite different. We folded and made the shapes, then designed our own nets which were made from at least three different 2D shapes.