IBM Ponder This #095 · March 2006

Puzzle for March 2006.

Consider 3 points chosen at random in an unit square. These points form a triangle. We want to compute the expected (average) area of this triangle. This can be done by answering the following questions.

1. Consider the minimum rectangle (with edges parallel to the edges of the unit square), R, containing the 3 random points. On average what is the area of R?

2. Given R, there are two configurations of the three points
   which occur with positive probability. They are

Case A - 2 points at opposite corners of R, remaining point in the
interior of R.

Case B - 1 point at a corner of R, 1 point along each of the 2 edges of R opposite the corner.

Show that the probabilities of occurrence of these two cases are independent of R. What are these probabilities?

3. What is the expected fraction of the area of R which is inside the triangle for each of the cases A and B above?

4. Combining the answers to questions 1-3, what is the expected area of the triangle formed by three points chosen at random in an unit square?

As usual we ask that you only submit your original work.

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The first 100 people who answer all 4 parts correctly will be listed. The answer will be posted a week after the 100th is received, or at the end of the month.

Solution

Best opened after a real attempt

To be added.