Why noughts and crosses is boring, how to make it more interesting, and why the first player should never lose (but not how).

Across mathematics you encounter situations where changing a 2 to a 3 turns a manageable problem into a completely untractable one.   I explain some examples of this phenomenon, then speculate wildly about what is going on.

Soap bubbles are round. Salt crystals are square. Why? What else?

There's a foolproof way of getting through to-do lists: put new items at the bottom, keep doing things from the top, and eventually everything will get done.  Of course, the real world is rarely that simple, and it can sometimes feel like we're battling a hydra, with two new heads growing for every one that we chop off.  But no matter how hopeless things seem, slow and steady progress can get us there in the end.

This was originally done in 4 minutes for an audience of postgraduate students, but could work at many lengths and levels.

Ultrafiilters are one answer to the question of how we can formalise "usually" in mathematics.  I explain the basics and give an application to Ramsey theory. 

This talk was for keen pre-university maths students, but a variant would work for a more general audience.