- Colin Wright's amazing talk on graph colouring, which started by asking us to complete a partially-completed 3-colouring of a small graph, and turned into a more-or-less complete proof, within a 5-minute talk, that there is no polynomial-time algorithm for 3-colouring a graph.
- Ross Atkins's talk about Braess's Paradox - a simple situation in which adding an extra road to a network, with no increase in traffic, results in longer average journey times. I should have known about this counter-intuitive result so I'm very glad to have found out about it, and especially with the wonderful demonstration with a network of springs that showed a mechanical realisation of the paradox.
- David Bedford's "What's my polynomial?" I love this because it is arguably what the late Raymond Smullyan called a "monkey trick". David asked you to think of a polynomial
*p*(*x*) with non-negative integer coefficients, and, for a single value of*x*of your choice, greater than any of the coefficients, tell him both*x*and*p*(*x*). He would then tell you your polynomial. Knowing that one needs*n*values to determine a polynomial of degree*n*, I was taken in by this!

I could have chosen many more examples: I'm certainly not ranking these presentations or any others. On another day I might have chosen a completely different set! But I'm certainly looking forward to coming across more wonderful mathematics this weekend!

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