S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
It is currently Tue, 21 May 2013 23:59:18 UTC

View unanswered posts | View active topics


Board index » Other Topics » Miscellaneous

All times are UTC [ DST ]



Post new topic Reply to topic  Page 1 of 1
  [ 6 posts ] 
  Print view Previous topic | Next topic 
Author Message
rdj5933mile5math64
 Post subject: Segments and Areas in a Unit Square
PostPosted: Mon, 11 Jun 2012 00:26:47 UTC 
Offline
Senior Member
User avatar

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
We are given a unit square and a finite number of line segments satisying the following conditions:

1. The total length of all of the segments is 18.

2. Each segment is parallel to one of the sides of the square.

3. The segments lie on the sides of the square and within the square (no part of any segment lies outside of the square).

The square is divided into region(s) by these line segments. Prove that one of these regions has an area that is greater than or equal to \frac{1}{100} units.

I have no idea how to formalize this...? Maybe I'm just being dumb, but I don't see a way to write up a pigeonhole principle argument, use direct method, or use contradiction?

_________________
math puns are the first sine of madness
-JDR
:mrgreen:
Top
 Profile  
 
outermeasure
 Post subject: Re: Segments and Areas in a Unit Square
PostPosted: Mon, 11 Jun 2012 03:51:37 UTC 
Offline
Moderator
User avatar

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
rdj5933mile5math64 wrote:
We are given a unit square and a finite number of line segments satisying the following conditions:

1. The total length of all of the segments is 18.

2. Each segment is parallel to one of the sides of the square.

3. The segments lie on the sides of the square and within the square (no part of any segment lies outside of the square).

The square is divided into region(s) by these line segments. Prove that one of these regions has an area that is greater than or equal to \frac{1}{100} units.

I have no idea how to formalize this...? Maybe I'm just being dumb, but I don't see a way to write up a pigeonhole principle argument, use direct method, or use contradiction?


Use a baby version of the isoperimetric inequality
Spoiler:
With a fixed perimeter and adjacent sides perpendicular, the figure with the biggest area is the square (Prove it!)

_________________
\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
Top
 Profile  
 
Soroban
 Post subject: Re: Segments and Areas in a Unit Square
PostPosted: Mon, 11 Jun 2012 16:48:11 UTC 
Offline
Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7949
Location: Lexington, MA
Hello, rdj5933mile5math64!

I have an intuitive approach . . .

Quote:
We are given a unit square and a finite number of line segments satisying the following conditions:

1. The total length of all of the segments is 18.

2. Each segment is parallel to one of the sides of the square.

3. The segments lie on the sides of the square and within the square (no part of any segment lies outside of the square).

The square is divided into region(s) by these line segments.
Prove that one of these regions has an area that is greater than or equal to \frac{1}{100} units.

Consider using 8 vertical lines and 10 horizontal lines, equally spaced.
The unit square is divided into (at most) 99 squares.
Hence, each square has area of at least \frac{1}{99}.

Consider using 9 vertical lines and 9 horizontal lines, equally spaced.
Then the unit square is divided into (at most) 100 squares.
Hence, each square has area of at least \frac{1}{100}

If the lines are not equally spaced,
. . some rectangles have an area less than \frac{1}{100}
. . and others have areas greater than \frac{1}{100}.
Top
 Profile  
 
outermeasure
 Post subject: Re: Segments and Areas in a Unit Square
PostPosted: Mon, 11 Jun 2012 17:05:42 UTC 
Offline
Moderator
User avatar

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Soroban wrote:
If the lines are not equally spaced,
some rectangles have an area less than \frac{1}{100}
and others have areas greater than \frac{1}{100}.


Err... nobody said the segments each have to span the entire width/height of the square, and moreover, you can have shapes other than rectangles.

_________________
\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
Top
 Profile  
 
Shadow
 Post subject: Re: Segments and Areas in a Unit Square
PostPosted: Mon, 11 Jun 2012 17:07:32 UTC 
Online
Moderator
User avatar

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
outermeasure wrote:
Soroban wrote:
If the lines are not equally spaced,
some rectangles have an area less than \frac{1}{100}
and others have areas greater than \frac{1}{100}.


Err... nobody said the segments each have to span the entire width/height of the square, and moreover, you can have shapes other than rectangles.


I think that was the point behind "intuitive". :)

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination
Top
 Profile  
 
rdj5933mile5math64
 Post subject: Re: Segments and Areas in a Unit Square
PostPosted: Tue, 12 Jun 2012 18:44:03 UTC 
Offline
Senior Member
User avatar

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
outermeasure wrote:
rdj5933mile5math64 wrote:
We are given a unit square and a finite number of line segments satisying the following conditions:

1. The total length of all of the segments is 18.

2. Each segment is parallel to one of the sides of the square.

3. The segments lie on the sides of the square and within the square (no part of any segment lies outside of the square).

The square is divided into region(s) by these line segments. Prove that one of these regions has an area that is greater than or equal to \frac{1}{100} units.

I have no idea how to formalize this...? Maybe I'm just being dumb, but I don't see a way to write up a pigeonhole principle argument, use direct method, or use contradiction?


Use a baby version of the isoperimetric inequality
Spoiler:
With a fixed perimeter and adjacent sides perpendicular, the figure with the biggest area is the square (Prove it!)


Got it! Thank you! :D

_________________
math puns are the first sine of madness
-JDR
:mrgreen:
Top
 Profile  
 
Display posts from previous:  Sort by  
Post new topic Reply to topic  Page 1 of 1
  [ 6 posts ] 

Board index » Other Topics » Miscellaneous

All times are UTC [ DST ]

Who is online

Users browsing this forum: No registered users

You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum

Search for:
Jump to:  
Contact Us | S.O.S. Mathematics Homepage
Privacy Statement | Search the "old" CyberBoard

users online during the last hour
Powered by phpBB © 2001, 2005-2011 phpBB Group.
Copyright © 1999-2013 MathMedics, LLC. All rights reserved.
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA