Next step: solve in three dimensions
August 30, 2011 7:48 AM Subscribe
Calculusfilter. A man is led to the center of a valuable field which he does not own. He coats his feet in blue paint so that his path can be traced. At dawn he begins walking. At a randomly selected time he will be told to stop walking, whereupon he will walk in a perfectly straight line back to the starting point. Then he will be given all the land that has been circumscribed by his blue path.
posted by foursentences to Grab Bag (89 answers total) 33 users marked this as a favorite
What path should the man walk to maximize his expected receipt of land?
Not for any kind of homework; just idle curiosity. More interested in the soluton, the shape and characteristics of the curve that results, than in the math by which you get there (though I'm somewhat interested in that, too). I'm envisioning a spiral with exponentially-increasing 'diameter' -- any chance it's the golden ratio spiral? How will he deal with the problem of overlap -- the fact that once his spiraling path gets large enough, his labors will begin 'adding' land he had already got on an earlier lap? Will he simply widen his diameter exponentially from the start, so that that he never begins overlapping at all? or will he accept some overlap?
(In case it turns out to be unsolvable if the time period is completely random, we could say that there's a maximum possible time period -- say, he knows he'll be walking for a random duration between 0 and 10 hours, and he knows his speed is exactly 1 mile per hour. Dunno if that helps nail down a solution?)
Thanks in advance!