Welcome to BrainDen.com - Brain Teasers Forum
|Welcome to BrainDen.com - Brain Teasers Forum. Like most online communities you must register to post in our community, but don't worry this is a simple free process. To be a part of BrainDen Forums you may create a new account or sign in if you already have an account.
As a member you could start new topics, reply to others, subscribe to topics/forums to get automatic updates, get your own profile and make new friends.
Of course, you can also enjoy our collection of amazing optical illusions and cool math games.
If you like our site, you may support us by simply clicking Google "+1" or Facebook "Like" buttons at the top.
If you have a website, we would appreciate a little link to BrainDen.
Thanks and enjoy the Den :-)
n-colored straight lines on painted plane
Posted 10 July 2013 - 04:00 PM
(1a) Paint the plane using 3 colors in such way that no 3-colored straight line exists.
(1b) Prove that if plane is painted with 4 colors, then a 3-colored straight line exists.
(2a) Paint the plane using as many colors as you can in such way that no 4-colored straight line exists.
(2b) Let m be the number of colors you used in (2a). Prove that if plane is painted with (m+1) colors, then a 4-colored straight line exists.
(3) What about 5-colored straight lines?
(4) What about n-colored straight lines?
Note: I don't have solutions to all of the problems above. Partial solutions are welcome.
Posted 10 July 2013 - 04:40 PM
Edited by Pickett, 10 July 2013 - 04:40 PM.
Posted 10 July 2013 - 04:51 PM
Spoiler for 1a
Edited by witzar, 10 July 2013 - 04:58 PM.
Posted 10 July 2013 - 06:16 PM
Spoiler for 1a
(1a) solvedSpoiler for
Edited by Pickett, 10 July 2013 - 06:16 PM.
Posted 10 July 2013 - 06:19 PM
Edited by HoustonHokie, 10 July 2013 - 06:21 PM.
Posted 10 July 2013 - 06:56 PM
Spoiler forSpoiler for 1b
While that's a good observation, it's not strictly a requirement.
Posted 10 July 2013 - 09:14 PM Best Answer
Edited by Anza Power, 10 July 2013 - 09:19 PM.
Posted 10 July 2013 - 09:16 PM
A more convincing proof of 1B.
Spoiler for 1B
(1b) solved. Well done.
0 user(s) are reading this topic
0 members, 0 guests, 0 anonymous users