wbp@hou2d.UUCP (W.PINEAULT) (12/18/84)
I thought that the point placing problem proposed by Peter Monta was *very* interesting and have spent a little time working on it. I also thought that Greg Rawlins interpretations of the problem all were wrong. To repeat the problem: Choose a sequence of points on the unit interval such that after choosing point n, there is exactly one point contained in each interval [j/n, (j+1)/n]. First comment: the conditions on whether the points are actually on the boundaries of the segments is not relavent to this note so I will represent everything as closed boundaries. In the limiting case this will matter as the relavent intervals approach having zero length. Second: The order of the first two points can be swapped so that the first point is in the interval [0,1/2]. Third: I have spent just a short time on this problem, but have a feel that there are an infinite number of arrangments of points to satisfy the problem, and that the points must be at irrational places (take that as you will)! This seems to be a combinatorial problem where the set of possibilities at each step increases faster than the deadends reached. A list of all possible places to put 4 points to satisfy the problem. The fourth point is assumed to be in the interval [0,1/2] since the problem is symmetric. The notation means: point number: [start, end] where start and end refer to all the posible placements of that point. 1: [0,1/4] 2: [1/2,2/3] 3: [3/4,1] 4: [1/4,1/2] 1: [0,1/4] 2: [3/4,1] 3: [1/2,2/3] 4: [1/4,1/2] 1: [1/4,1/3] 2: [1/2,3/4] 3: [3/4,1] 4: [0,1/4] 1: [1/4,1/3] 2: [2/3,1] 3: [1/3,2/3] 4: [0,1/4] Wayne Pineault AT&T Consumer Products "No matter where you go, well..., there you are!"