Ideas for Math Pojects

I've included a few ideas for math projects that allows students to investigate a problem. These problems are not run of mill and hence they will provide students a glimpse of how mathematical research can is done. This list is certainly not exhaustive. It is targeted to juniors only. The ideas are grouped according to major topics in mathematics.

Arithmetic/Number Theory
There are many sets of interesting numbers. Examples include polite numbers, amicable numbers and frugal numbers. Come up with your own set of numbers with some special properties and investigate.


Combinatorics
How many latin squares can be formed using a 3 by 3 grid? What about an n by n grid?
What is the answer to the 36 officer problem?

Geometry
Investigate the tiling of polygons. What polygons can be tiled and which can't?
Investigate the ways of estimating the volume/surface area of irregularly shaped object

Applied mathematics
What is the most efficient way to insert people in a duty roster whom have their own preferences?
Do you get more wet when you run in the rain or when you walk?
What is the fastest and cheapest way to get to a location in Singapore (for instance home to school) using MRT, Train and Taxi?
How do we program a lift so that it can transport people efficiently?
How can we estimate the volume of the Earth?
Why do buses sometimes come in pairs (or threes)?