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April 2022 - Challenge
Minimizing a sum for two orthogonal vectors
This month’s puzzle is based on a suggestion by Marco Bellocchi. Thanks, Marco!
Let be two vectors, and define the function
We wish to minimize
, subject to the following constraints on
: The vectors need to be orthogonal to each other and to the vector
and of norm 1, i.e.,
1.
2.
3.
The lower bound for appears to be 4, but we know of no proof for general
. Nevertheless, we refer to a pair
giving
as being an "optimal solution".
For example, suppose and
x = [-0.3562014 , 0.63245526, -0.3688259 , -0.36163773, 0.45420977]
y = [-0.03997346, -0.6324558 , -0.05259805, -0.04540969, 0.77043701]
Then are very close to being an optimal solution, up to a small error of at most 0.0000001
Your goal For , find a pair of
that are 0.0000001-close to being an optimal solution. Give your solution in the exact format given above.
A bonus "*" will be given for finding an **exact** solution ( ) where
are both rational vectors (
), for some
at least 5. Give your solution in a format similar to that above, but with numbers written as "a/b", e.g.,
x = [1/2, 1/3, 1/6]
A bonus "**" will be given for finding an exact rational solution for any , or for proving 4 is indeed the minimum value of
. (We do not know of such a solution or such a proof.)
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:
29/03/2022 @ 14:00 PM EST
Solution:
04/05/2022 @ 12:00 PM EST
List Updated:
11/05/2022 @ 13:45 PM EST
People who answered correctly:
**Bert Dobbelaere (30/3/2022 9:34 PM IDT)
**Anders Kaseorg (31/3/2022 1:29 AM IDT)
Richard Gosiorovsky (31/3/2022 5:19 PM IDT)
**Bertram Felgenhauer (1/4/2022 7:43 AM IDT)
Kerry Soileau (1/4/2022 3:47 PM IDT)
Graham Hemsley (1/4/2022 5:47 PM IDT)
Joerg Schepers (1/4/2022 8:43 PM IDT)
**Phil Proudman (2/4/2022 9:03 PM IDT)
**Laurent Lessard (3/4/2022 7:58 AM IDT)
Daniel Chong Jyh Tar (3/4/2022 1:44 PM IDT)
Balakrishnan V (4/4/2022 2:27 AM IDT)
**Uoti Urpala (4/4/2022 2:55 AM IDT)
Gary M. Gerken (4/4/2022 3:28 AM IDT)
Sachal Mahajan (4/4/2022 6:20 AM IDT)
Lorenz Reichel (5/4/2022 2:56 PM IDT)
**Alper Halbutogullari &Ali Ekber Gurel (6/4/2022 9:38 AM IDT)
*Evert van Dijken (6/4/2022 12:04 PM IDT)
Paul Revenant (7/4/2022 5:10 PM IDT)
Stephen Barr (12/4/2022 7:15 AM IDT)
Fabio Michele Negroni (13/4/2022 1:15 PM IDT)
**David Greer (14/4/2022 2:32 PM IDT)
**Vladimir Volevich (14/4/2022 8:10 PM IDT)
**Nyles Heise (15/4/2022 8:58 AM IDT)
Clive Tong (16/4/2022 11:14 AM IDT)
Harald Bögeholz (17/4/2022 5:25 AM IDT)
Tamir Ganor & Shouky Dan (17/4/2022 5:15 PM IDT)
John Goh (18/4/2022 5:13 AM IDT)
**Oleg Vlasii (19/4/2022 6:32 PM IDT)
Dan Dima (21/4/2022 8:12 PM IDT)
*Reda Kebbaj (23/4/2022 2:07 AM IDT)
Sri Mallikarjun J (23/4/2022 10:08 AM IDT)
**Latchezar Christov (23/4/2022 2:05 PM IDT)
**Aaron Kim (23/4/2022 4:55 PM IDT)
**Elliot Hong (23/4/2022 5:54 PM IDT)
**Grace Park (24/4/2022 4:53 PM IDT)
Teawoo Kim (24/4/2022 5:58 PM IDT)
Nir Lavee & Omer Shechter (24/4/2022 6:23 PM IDT)
**Joon Park (25/4/2022 5:38 AM IDT)
**Keun Hyong Kwak (25/4/2022 7:23 AM IDT)
Andreas Knüpfer (25/4/2022 6:32 PM IDT)
Kang Jin Cho (25/4/2022 8:13 PM IDT)
**Austin Lee (25/4/2022 9:17 PM IDT)
Chris Shannon (26/4/2022 8:26 PM IDT)
Sebastian Bohm & Martí Bosch (27/4/2022 11:43 PM IDT)
Daniel Bitin (29/4/2022 12:57 PM IDT)
**Motty Porat (30/4/2022 12:40 AM IDT)
Rob Pratt (30/4/2022 1:03 AM IDT)
Alexander Gramolin (30/4/2022 7:36 AM IDT)
Albert Stadler (30/4/2022 10:42 AM IDT)
Tim Walters (1/5/2022 1:08 AMIDT)
**Radu-Alexandru Todor (1/5/2022 3:03 AM IDT)
Amos Guler (2/5/2022 8:54 PM IDT)
**Li Li (3/5/2022 7:10 AM IDT)