So here I am on the campus of the University of Warwick, at the end of the first day of the 25th British Combinatorial Conference.
Two things struck me right away. First, I really do feel that this is my extended family; I have so many good friends here. And on the other hand, it is a conference at which I never really feel off duty. As well as my constitutional jobs of chairing the conference business meeting and (if they re-elect me) the committee meeting, I have by tradition the job of running the problem sessions (we are having two at this meeting), doing a turn at the conference concert, making a speech at the dinner (if I don’t succeed in avoiding this), and various announcements.
Anyway, today I only had my talk, the first problem session, and a short announcement. Oh, and there was a piece of business which the committee had collectively forgotten; a completely unrelated remark at the reception threw a switch in my brain, and I had to go around the reception sorting it out.
Apart from all that, a very good collection of talks. Since I am rather busy, I will be brief, and simply report a small gem from Gil Kalai’s talk.
This concerns Buffon’s needle: if we throw a needle of unit length randomly onto a floor with parallel lines one unit apart, what is the probability that the needle crosses a line? Here is the solution, as Gil described it.
- First, we can replace “probability it crosses a line” by “expected number of lines crossed”, since the probability of crossing a number of lines different from 0 or 1 is zero.
- Next, we replace “Buffon’s needle” by “Buffon’s noodle”, a wiggly (but smooth) line of length l. (For technical reasons it is better to replace it first by a polygonal arc, and then extend the results to smooth curves by a limiting process.)
- By concatenating two noodles of lengths l1 and l2, and using the linearity of expectation, we see that the expected number of lines crossed is a linear function of the length, say cl for some constant c; our job is to find c.
- Now consider the case where the noodle is a circle of diameter 1 (and so length π). The expected number of lines crossed in this case is clearly 2; so the constant c is equal to 2/π.
- So in the original needle problem, the answer is 2/π.
A final remark. When I arrived at my accommodation, I was given a leaflet about connecting to Warwick guest wi-fi, which implied that without a mobile phone it was impossible to connect. The fact that you are reading this now might suggest that I have succumbed and got a phone; but in fact the answer is simpler, they have eduroam, which works perfectly (although, after my computer went down, eduroam was the very last thing I managed to restore). So I am still a dinosaur!