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# n-colored straight lines on painted plane

Best Answer Anza Power, 10 July 2013 - 09:14 PM

Spoiler for 2a 2b 3 4

Go to the full post

10 replies to this topic

### #1 witzar

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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.

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### #2 Pickett

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Posted 10 July 2013 - 04:40 PM

Spoiler for 1a

Edited by Pickett, 10 July 2013 - 04:40 PM.

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### #3 witzar

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Posted 10 July 2013 - 04:51 PM

Spoiler for 1a

(1a) solved

Spoiler for

Edited by witzar, 10 July 2013 - 04:58 PM.

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### #4 Pickett

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Posted 10 July 2013 - 06:16 PM

Spoiler for 1a

(1a) solved

Spoiler for

Spoiler for hm...

Edited by Pickett, 10 July 2013 - 06:16 PM.

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### #5 HoustonHokie

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Posted 10 July 2013 - 06:19 PM

(1a) solved

Spoiler for

Spoiler for 1b

Edited by HoustonHokie, 10 July 2013 - 06:21 PM.

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### #6 HoustonHokie

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Posted 10 July 2013 - 06:53 PM

Spoiler for 2a

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### #7 k-man

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Posted 10 July 2013 - 06:56 PM

(1a) solved

Spoiler for

Spoiler for 1b

While that's a good observation, it's not strictly a requirement.

Spoiler for alternative 1a solution

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### #8 Anza Power

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Posted 10 July 2013 - 09:04 PM

A more convincing proof of 1B.

Spoiler for 1B

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### #9 Anza Power

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Posted 10 July 2013 - 09:14 PM   Best Answer

Spoiler for 2a 2b 3 4

Edited by Anza Power, 10 July 2013 - 09:19 PM.

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### #10 witzar

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Posted 10 July 2013 - 09:16 PM

A more convincing proof of 1B.

Spoiler for 1B

(1b) solved. Well done.

Spoiler for minor comment

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