IBM Research | Ponder This | December 2004 challenges

# Ponder This

## December 2004

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Ponder This Challenge:

Puzzle for December 2004

This month's puzzle is proposed by Michael Brand.

Consider N vectors, V_1 ... V_N, (not all zero) in an N-dimensional space. Project these vectors orthogonally to a 2-dimensional plane, P. Let us denote the results for the i'th vector as (a_i,b_i) in Cartesian coordinates defined on the plane. We will be interested in considering these coordinates as a complex number: x_i = a_i + j*b_i.

Now, these N vectors hold the following property: regardless of the chosen P, the sum of all {x_i}^2 is always equal to 0.

The question: give an explicit description of the property that the N vectors most hold. Prove that this property is both sufficient and necessary.

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: 12/01/2004 @ 9:00 AM ET
Solution: 01/01/2005 @ 8:30 AM ET
List Updated: 12/01/2004 @ 9:00 AM ET

### People who answered correctly:

Dan Dima (12.01.2004 @10:28:45 AM EST)
Raicho Tchalkov (12.01.2004 @11:52:08 AM EST)
Stephen Lesko (12.01.2004 @03:35:49 PM EST)
Rubén Hernández Aza (12.01.2004 @04:04:32 PM EST)
Divyesh Dixit (12.01.2004 @04:30:23 PM EST)
Frank Mullin (12.01.2004 @04:41:18 PM EST)
Dion K Harmon (12.01.2004 @08:01:40 PM EST)
Patrick J. LoPresti (12.01.2004 @11:22:38 PM EST)
Yong Liu (12.02.2004 @09:27:34 AM EST)
Vineet Dwivedi (12.02.2004 @11:48:15 AM EST)
Kalyana Chakravarthy (12.02.2004 @01:34:03 PM EST)
Max Alekseyev (12.02.2004 @06:59:45 PM EST)
Vineet Gupta (12.02.2004 @08:17:23 PM EST)
Piotr Zielinski (12.02.2004 @09:08:06 PM EST)
Eugene Vasilchenko (12.03.2004 @01:05:16 AM EST)
Amit Jayant Deshpande (12.03.2004 @01:05:16 AM EST)
John Ritson (12.03.2004 @05:29:52 AM EST)
Mayank Kabra (12.03.2004 @08:50:23 AM EST)
Luke Pebody (12.03.2004 @12:50:23 AM EST)
Hao Wu (12.03.2004 @01:07:25 PM EST)
Golovach Ivan (12.03.2004 @01:41:39 PM EST)
Lauri Ylinen (12.03.2004 @03:39:03 PM EST)
Alexandru Sofronia (12.04.2004 @04:42:55 PM EST)
Hagen von Eitzen (12.04.2004 @05:33:34 PM EST)
Vanhulle Christian (12.04.2004 @06:15:53 PM EST)
Erick Wong (12.05.2004 @02:19:51 AM EST)
Kevin Costello (12.05.2004 @07:05:44 AM EST)
Hoffmann Klaus (12.05.2004 @11:16:29 AM EST)
John G. Fletcher (12.05.2004 @07:45:56 PM EST)
Avi Gozolchiani (12.06.2004 @11:22:00 AM EST)
Zabara Maxim (12.06.2004 @09:26:39 AM EST)
Yakov Sirotkin (12.06.2004 @03:42:23 PM EST)
Kim Se Kwon (12.06.2004 @08:23:53 PM EST)
Dharmadeep Muppalla (12.08.2004 @05:35:13 AM EST)
Devin Myers (12.08.2004 @10:44:01 AM EST)
Claudio Baiocchi (12.09.2004 @07:04:51 AM EST)
Jose H. Nieto (12.10.2004 @10:50:14 PM EST)
John Tromp and Troy Lee (12.13.2004 @03:48:41 AM EST)
Sergey Povetkin (12.15.2004 @11:33:57 AM EST)
Alan O'Donnell (12.15.2004 @05:14:36 PM EST)
Michael Nedzelsky (12.16.2004 @05:44:20 PM EST)
Andreas Kabel (12.17.2004 @07:48:11 PM EST)
Chris Hills (12.20.2004 @09:38:19 AM EST)
Siddhartha Kanungo (12.22.2004 @11:55:22 AM EST)
Leonardo Liang (12.24.2004 @04:32:22 AM EST)
Dr. Bert Fischer (12.24.2004 @05:46:19 AM EST)
Nagi Nahas (12.24.2004 @03:07:43 PM EST)
Mihai Cioc (12.24.2004 @06:55:23 PM EST)
Alex Tikhonov (12.28.2004 @09:29:46 AM EST)
Ananda Raidu (12.31.2004 @06:56:46 AM EST)
IQSTAR (12.31.2004 @11:42:38 AM EST)
Jonathan Poritz (12.31.2004 @02:18:00 PM EST)
Aditya K. Prasad (12.31.2004 @07:48:28 PM EST)
Ramanjit Ahuja (01.03.2005 @11:10:55 AM EST)

Attention: If your name is posted here and you wish it removed please send email to the ponder@il.ibm.com.