I am teaching a "computation" course, where we get students to explore mathematics and statistics via a high-level programming language. So the course is less about mathematics proper, and more about computation, and the process of learning about mathematics.
I am looking for good examples of the form where the main ideas behind the techniques are easy to motivate, and yet the technique (aided with a computer) produces otherwise difficult-to-obtain results.
I've been hoping to find a good example from the MCMC family of ideas. Perhaps it's because I'm not terribly familiar with this branch of stats myself that I haven't developed good basic examples.
Perhaps this is a stretch, but in these family of ideas would there be a good, simple way of estimating the volume of an icosahedron?
I don't need examples quite that sophisticated. But I would like the examples to be intuitive-enough to stand on their own, but flexible enough so the student could have some idea of how to apply the technique to a wide variety of problems.
These students know linear algebra and multi-variable calculus, and a little bit about programming languages.