IBM Research | Ponder This | March 2006 challenges

# Ponder This

## March 2006

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Ponder This Challenge:

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?

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.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Challenge: 03/01/2006 @ 09:00 AM EST
Solution: 03/11/2006 @ 12:00 PM EST
List Updated: 03/01/2006 @ 09:00 AM EST

Roberto Tauraso (03.01.2006 @11:58:25 AM EDT)
Joseph DeVincentis (03.01.2006 @01:41:26 PM EDT)
William C Hasenplaugh (03.01.2006 @03:50:13 PM EDT)
Balakrishnan V (03.01.2006 @06:21:05 PM EDT)
Arthur Breitman (03.01.2006 @07:31:40 PM EDT)
Aditya Mahajan (03.01.2006 @05:53:20 PM EDT)
Yurun Liu (03.01.2006 @10:04:39 PM EDT)
Jing Shan (03.02.2006 @12:30:56 AM EDT)
Frank Yang (03.02.2006 @12:54:56 AM EDT)
Lukas Saul (03.02.2006 @01:53:17 AM EDT)
Malcolm J. Powell (03.02.2006 @12:43:38 PM EDT)
Michael Swart (03.02.2006 @01:15:27 PM EDT)
Greg Bubnis (03.02.2006 @01:46:41 PM EDT)
Frank Mullin (03.02.2006 @02:29:45 PM EDT)
Dan Dima (03.03.2006 @03:03:29 AM EDT)
Se Kwon Kim (03.03.2006 @10:18:06 AM EDT)
Manuel Gräf (03.03.2006 @10:29:45 AM EDT)
John G. Fletcher (03.03.2006 @01:55:09 PM EDT)
aman_cc (03.03.2006 @05:18:41 PM EDT)
Lovro Puzar (03.03.2006 @06:20:58 PM EDT)
Graeme McRae (03.03.2006 @08:45:53 PM EDT)
John Dalbec (03.04.2006 @12:24:06 AM EDT)
Sandeep Rathour (03.04.2006 @01:36:33 AM EDT)
Jacques Willekens (03.04.2006 @05:40:41 AM EDT)
Allan Lazarovici (03.04.2006 @02:26:31 PM EDT)
Chris Messer (03.05.2006 @05:09:27 PM EDT)
David McQuillan (03.05.2006 @06:49:07 PM EDT)
James Dow Allen (03.05.2006 @11:57:47 PM EDT)
Gaurav Agrawal (03.06.2006 @01:37:53 AM EDT)
Ed Sheppard (03.06.2006 @11:28:43 AM EDT)
Phil Muhm (03.06.2006 @10:22:03 PM EDT)
Vance Berger and Brendan Berger (03.07.2006 @12:41:54 PM EDT)
David Friedman (03.08.2006 @11:01:17 AM EDT)
Bill Schwennicke (03.08.2006 @05:42:01 PM EDT)
Vincent Vermaut (03.09.2006 @02:34:38 AM EDT)
David Wang (03.09.2006 @12:09:08 PM EDT)
Rajkumar Pal (03.09.2006 @07:18:31 PM EDT)
Zhou Guang (03.09.2006 @08:02:16 PM EDT)
Manikandan Jagadeesan (03.10.2006 @02:08:46 PM EDT)
Mark Pilloff (03.10.2006 @06:02:50 PM EDT)
Wolf Mosle (03.11.2006 @07:36:58 PM EDT)
John Ritson (03.11.2006 @11:19:27 PM EDT)
Aliekber Gurel (03.12.2006 @01:32:04 AM EDT)
Dion K Harmon (03.12.2006 @11:01:32 AM EDT)
Joaquim N. Carrapa (03.12.2006 @08:31:45 PM EDT)
Pascal Zimmer (03.13.2006 @06:33:15 PM EDT)
Rubén Hernández Aza (03.13.2006 @10:48:40 PM EDT)
Immanuel Litzroth (03.14.2006 @10:29:50 AM EDT)
Evgeny (03.14.2006 @10:22:59 PM EDT)
Ariel Flat (03.15.2006 @03:25:28 PM EDT)
Michael Szydlo (03.15.2006 @03:44:07 PM EDT)
Job Philip (03.17.2006 @06:55:45 AM EDT)
Du Yang (03.17.2006 @10:05:11 PM EDT)
Kennan Shelton (03.20.2006 @10:19:15 PM EDT)
Adel El-Atawy (03.20.2006 @11:33:58 PM EDT)
Harinarayanan E V (03.21.2006 @06:20:52 AM EDT)
Alexey Vorobyov (03.22.2006 @07:46:41 PM EDT)
Tom Sirgedas (03.23.2006 @11:59:35 AM EDT)
Ranchu Mathew (03.23.2006 @12:09:48 PM EDT)