David Corfield’s Towards a Philosophy of Real Mathematics spends much of its time on plausible reasoning within mathematics. He mentions a number of inductive “leaps of faith” including John MacKay’s observation about 196884 appearing both as a coefficient in the Fourier expansion of the j function and (almost) as the dimension of a representation of the monster group. That coincidence turned out to be no coincidence. He also mentions Fermat’s less successful conjecture that all Fermat numbers are prime.
One such extrapolation which I have only come across once concerns points irregularly placed around a circle. I don’t know whether this is common knowledge in mathematical circles but it certainly wasn’t common knowledge to me when my daughter, then in one of the lower forms of secondary school, brought it home. The problem was simple:
N points are placed irregularly on the circumference of a circle. Straight lines are drawn between each pair of points. Into how many pieces is the circle divided?
So, we started with N=1 and found that the circle was still in one bit. f(1) = 1.
With N=2 the circle is clearly divided into two. f(2) = 2.
It’s still pretty easy to see that f(3) = 4 and f(4) = 8 but after that the diagram gets a bit crowded. But the pattern was emerging fast. I told my daughter to check that f(5) = 16 and she did. It is.
At this point the homework exercise was effectively finished: 1, 2, 4, 8, 16 form a fairly predictable sequence and when she came back and told me that f(6) = 33 I told her to go away and draw the picture more carefully. She did and still found f(6) to be 33. I was now interested again and, after a lot of work, we found the quartic that just happens to return
1, 2, 4, 8, 16, 33
for its first few terms. Was the conjecture I made less natural than Mackay’s?