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The integers from 1 to 6 are paired at random, each pair being regarded as the endpoints of a random interval. What is the probability that among these 3 intervals there is one that meets all the others?
If that is too easy, try the case of pairings from 1 to 8. And if that is too easy, try to generalize to any n.
AddendumThis problem originated in the paper: "Random Intervals" by Justicz, Scheinerman, and Winkler, American Math Monthly 97 (1990) 881-889.
They proved that a certain limiting probability equalled 2/3 and then: "It was when the authors asked themselves how fast the probability converged to 2/3 that the startling truth emerged: The probability that in a collection of n random intervals there is one which intersects all the others is exactly 2/3."
AN ANNOUNCEMENT FROM STAN WAGONToroidal Trousers on the Web
Dan Schwalbe and I, inspired by Helaman Ferguson, now have a pretty nifty movie that goes from a triply punctured torus to the Costa surface. Full details will be in a forthcoming article in "Mathematica in Education and Research"; editor David Fowler has put a Java-based version of the movie on the web. Others have made such movies before, but it is remarkable that it con be done in Mathematica with some very simple formulas. In particular, we get a very good-looking deformation by simply taking the linear combination: k (Costa) + (1 - k) (Torus).
The URL is:
And Dan has put some other relevant things up on his web page:
© Copyright 1999 Rob Leduc. Reproduced with permission.