IBM Research | Ponder This | February 2005 challenges

# Ponder This

## February 2005

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Ponder This Challenge:

Puzzle for February 2005

Here's a geometric puzzle. All three parts need to be solved.

Start with a right triangle T1 whose vertices are (0,0), (0,1), (1,1). The angle at (0,0) is /4. Use the hypotenuse of T1 as the leg of a new triangle T2, whose angle at (0,0) is /8, and which extends down towards the x-axis. Use the hypotenuse of T2 as the leg of a new triangle T3, whose angle at (0,0) is /16. Continue ad infinitum.

The angles at (0,0) form an infinite series which sums to /2, so the limit of the process is a line segment on the x-axis.

Part 1: How long is that segment?

Part 2: Find an equation in polar coordinates (r,) of a curve passing through all the vertices of the resulting triangles, except possibly (0,0).

Part 3: Find an equation for that curve in rectangular coordinates (x,y).

The first 100 people who answer 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: 01/31/2005 @ 12:00 PM EST
Solution: 03/01/2005 @ 08:30 AM EST
List Updated: 01/31/2005 @ 08:30 AM EST

Tomas G. Rokicki (01.31.2005 @01:36:38 PM EDT)
Michael Nedzelsky (01.31.2005 @05:01:19 PM EDT)
Max Alekseyev (01.31.2005 @06:45:03 PM EDT)
Tamas Visegrady (01.31.2005 @07:06:33 PM EDT)
Eugene Vasilchenko (01.31.2005 @07:15:14 PM EDT)
Mayank Kabra (02.01.2005 @12:36:33 AM EDT)
Rustem Aidagulov (02.01.2005 @01:05:30 AM EDT)
IQSTAR (02.01.2005 @02:01:48 AM EDT)
Narendra Agrawal (02.01.2005 @04:40:40 AM EDT)
Dan Dima (02.01.2005 @07:14:10 AM EDT)
Raicho Tchalkov (02.01.2005 @07:16:09 AM EDT)
Kenneth Chan (02.01.2005 @07:16:10 AM EDT)
Jose H. Nieto (02.01.2005 @08:41:31 AM EDT)
Ivan Golovach (02.01.2005 @10:25:31 AM EDT)
Joe BGI SF Fendel (02.01.2005 @01:20:54 PM EDT)
Dinesh Layek (02.01.2005 @03:50:44 AM EDT)
Dion K Harmon (02.01.2005 @04:00:45 PM EDT)
Leonardo Liang (02.02.2005 @02:06:22 AM EDT)
Hao Wu (02.02.2005 @02:20:51 AM EDT)
Dharmadeep Muppalla (02.02.2005 @03:46:09 AM EDT)
Igor Malashkin (02.02.2005 @03:47:33 AM EDT)
Dr. Luke Pebody (02.02.2005 @05:19:51 AM EDT)
Guangzhong Sun (02.02.2005 @06:35:19 AM EDT)
Vineet Dwivedi (02.02.2005 @12:43:06 PM EDT)
Yunpeng Zhou (02.02.2005 @12:51:46 PM EDT)
Hagen von Eitzen (02.02.2005 @03:55:31 PM EDT)
Vineet Gupta (02.02.2005 @04:12:14 PM EDT)
Vincent Vermaut (02.02.2005 @04:29:12 PM EDT)
Ken Crounse (02.02.2005 @04:49:03 PM EDT)
Ashwin Lall (02.02.2005 @05:12:59 PM EDT)
Piros Lucian (02.02.2005 @07:29:47 PM EDT)
Kennedy (02.02.2005 @08:00:30 PM EDT)
Kalyana Chakravarthy (02.02.2005 @08:30:19 PM EDT)
Bart De Vylder (02.03.2005 @07:15:19 AM EDT)
David Friedman (02.03.2005 @11:53:43 AM EDT)
Christian Vanhulle (02.03.2005 @12:40:35 PM EDT)
Vinko Marinkovic (02.03.2005 @09:22:00 AM EDT)
Sachin Shukla (02.04.2005 @03:51:45 AM EDT)
Victor Chang (02.04.2005 @04:16:37 AM EDT)
Omer Faruk Alis (02.04.2005 @04:47:36 AM EDT)
Dipak K Jha (02.05.2005 @12:47:33 PM EDT)
Pascal Brisset (02.04.2005 @09:53:25 AM EDT)
Tameen Khan (02.04.2005 @11:03:22 AM EDT)
Amit Jain (02.04.2005 @02:13:00 PM EDT)
Emma Rebato (02.04.2005 @02:22:22 PM EDT)
Anand (02.04.2005 @03:18:57 PM EDT)
John Hart (02.04.2005 @04:31:50 PM EDT)
Daniel Bitin (02.04.2005 @04:43:48 PM EDT)
George Ciobanu (02.04.2005 @06:28:35 PM EDT)
Samir Kagadkar (02.04.2005 @06:39:36 PM EDT)
Coserea Gheorghe (02.04.2005 @07:38:29 PM EDT)
Alexey Vorobyov (02.04.2005 @08:33:04 PM EDT)
Kim Se Kwon (02.04.2005 @11:31:38 PM EDT)
Dave Dodson (02.04.2005 @11:54:30 PM EDT)
Rubén Hernández Aza (02.05.2005 @07:30:53 AM EDT)
Lizhao Zhang (02.05.2005 @06:19:48 PM EDT)
Pradheep Elango (02.06.2005 @01:40:55 AM EDT)
Oskar Sigvardsson (02.06.2005 @07:07:47 PM EDT)
Sunil Nandihalli (02.07.2005 @12:12:20 AM EDT)
Shubin Hu (02.07.2005 @01:17:16 AM EDT)
John G. Fletcher (02.07.2005 @01:22:58 AM EDT)
Arup Salui (02.07.2005 @02:08:26 AM EDT)
John Tromp (02.07.2005 @10:02:57 AM EDT)
Scott Hotes (02.07.2005 @03:45:58 PM EDT)
Dave Biggar (02.08.2005 @05:09:00 AM EDT)
Guler Amos (02.08.2005 @10:23:57 AM EDT)
Claudio Baiocchi (02.08.2005 @12:04:38 PM EDT)
Natalia Kolganova (02.08.2005 @02:44:00 PM EDT)
Raghav Bhaskar (02.09.2005 @10:02:22 AM EDT)
Caltech Math Students (02.09.2005 @04:40:56 PM EDT)
John Ritson (02.09.2005 @10:58:28 PM EDT)
Thiru Sampath (02.09.2005 @11:02:00 PM EDT)
Divyesh Dixit (02.10.2005 @05:35:53 AM EDT)
Steve Powell (02.10.2005 @08:15:45 AM EDT)
Maxim Zabara (02.11.2005 @02:32:23 AM EDT)
Emile Joubert (02.11.2005 @10:53:42 AM EDT)
Vince Lynch (02.11.2005 @01:14:18 PM EDT)
David McQuillan (02.11.2005 @05:28:42 PM EDT)
Florbela Isabel Carrasqueiras (02.12.2005 @01:10:21 PM EDT)
Bill Webb (02.12.2005 @02:55:24 PM EDT)
Boris Silantiev (02.12.2005 @04:33:29 PM EDT)
Daniel Chong Jyh Tar (02.12.2005 @11:18:18 PM EDT)
Herbert Groscot (02.13.2005 @08:05:32 AM EDT)
Lovro Puzar (02.13.2005 @01:15:37 PM EDT)
Karl D'Souza (02.13.2005 @05:33:51 PM EDT)
Franco Bassi (02.13.2005 @06:19:41 PM EDT)
Tim Lewis (02.16.2005 @04:57:36 PM EDT)
Nathaniel Litton (02.17.2005 @01:34:00 PM EDT)
Aditya Utturwar (02.18.2005 @10:02:43 AM EDT)
Narayanan Rajamanin (02.19.2005 @07:08:19 PM EDT)
Sergey Povetkin (02.20.2005 @01:40:21 PM EDT)
Jay Lewandowski (02.21.2005 @01:18:48 AM EDT)
Ajay Agrawal (02.24.2005 @05:16:08 AM EDT)
Richard A Bauer (02.24.2005 @09:28:58 PM EDT)
John Dalbec (02.26.2005 @12:06:55 AM EDT)
Yakov Sirotkin (02.26.2005 @03:26:05 AM EDT)
Sridevi Parise & Naval Verma (02.28.2005 @02:28:07 AM EDT)
Petru Budur (02.28.2005 @05:43:08 AM EDT)

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