December 16, 1999
OLD CHALLENGE. How fast do you get home with a random walk on the line? in the plane? in three-space? in n-space?
ANSWER. In his winning response, Eric Brahinsky reports that the median number of steps to get home is 2-4 on the line, about 32 in the plane, and infinite in higher dimensions. Indeed, on the line, you have a 50 percent chance of retracing your first, departure step to return home in 2 steps, and a 50 percent chance of requiring 4 or more steps. (Of course the total number of steps home must be even.)
For the plane, Brahinsky reports that computer simulations suggest a median of about 32 steps. In three-space, you are no longer 100 percent sure of returning home; in fact, the probability of ever returning home is less than 50 percent (about 34 percent). Therefore the median is infinite. And it's only worse in higher dimensions. So maybe we're fortunate to live on the Earth's surface (two-dimensional), or we'd probably all be lost by now.
NEW CHALLENGE. Justin Smith calls 5939 a "right" prime because it remains prime after dropping any number of digits from the right: 5939, 593, 59, and 5 are all prime. How many right primes are there less than 1000? Is there a largest right prime?
Send answers, comments, and new questions by email to Frank.Morgan@williams.edu, to be eligible for Flatland and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan's homepage is at www.williams.edu/Mathematics/fmorgan.
Math Chat wishes all its readers a happy season and progressive new year.
Copyright 1999, Frank Morgan.