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Problem of the Week 945

A Golfing Dilemma

A group of 16 people wishes to play 7 rounds of golf in foursomes; that is, each round consists of four groups of four playing the course. They want to arrange things so that each pair plays together at least once in a foursome. Design a scheme to arrange this.

Notes: This is a real-world problem suggested by a (non-math) colleague, Mark Davis. Mark conjectures that if the number of golfers is n, divisible by 4, then the number of rounds needed so that everyone plays with everyone else is (n/2) - 1. But we have no proof. Naturally, we also wonder if this problem falls within the purview of well-known methods in combinatorial design. Comments welcome.

Interesting Bicycle Work

Recall the famous Sherlock Holmes problem of determining the direction of travel of a bike given the tracks in mud of its rear and front tires. David Finn (Rose-Hulman) has found bike tracks that are ambiguous in several ways:

  1. A single nonlinear track that could be made either by a bike or by a unicycle.
  2. A pair of nonlinear tracks that could have been made by a bike traveling left-right, or by a bike traveling right-left.

Images of both are available on his web page. While some theoretical aspects of the construction need to be hammered out, it all appears to work, and I think these ideas are quite fascinating, especially as the Sherlock Holmes bike problem has been around for quite a long time (e.g., in the Konhauser-Velleman-Wagon problem book, "Which Way Did the Bicycle Go?").

The relevant URLs are as follows. I have used Mathematica to generate more of the unicycle/bike ambiguous track.
© Copyright 2001 Stan Wagon. Reproduced with permission.

30 October 2001