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 Post subject: Circles and Squares in RectanglesPosted: Thu, 24 May 2012 18:11:34 UTC
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Joined: Wed, 4 Apr 2012 03:51:40 UTC
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Location: Hockeytown aka Detroit
Given a 20x25 rectangle, insert 120 unit squares in it. Prove that one can place a circle with unit diameter in the rectangle, such that it doesn't intersect any of the squares.

This is a "there exists" problem. Therefore, I want to say that this problem is pigeonhole principle or probabilistic method. Hmmmm probabilistic method seems a little difficult... But I have no idea how pigeonhole principle works here either. Unfortunately, I can't really try small numbers either since the numbers 20,25,120 show no real pattern.

So, in short I have no idea as to how I should approach this problem.~

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 Post subject: Re: Circles and Squares in RectanglesPosted: Thu, 24 May 2012 18:22:39 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6005
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
rdj5933mile5math64 wrote:
Given a 20x25 rectangle, insert 120 unit squares in it. Prove that one can place a circle with unit diameter in the rectangle, such that it doesn't intersect any of the squares.

This is a "there exists" problem. Therefore, I want to say that this problem is pigeonhole principle or probabilistic method. Hmmmm probabilistic method seems a little difficult... But I have no idea how pigeonhole principle works here either. Unfortunately, I can't really try small numbers either since the numbers 20,25,120 show no real pattern.

So, in short I have no idea as to how I should approach this problem.~

Looks like the question is using or something like that.

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 Post subject: Re: Circles and Squares in RectanglesPosted: Fri, 25 May 2012 05:01:17 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
outermeasure wrote:
rdj5933mile5math64 wrote:
Given a 20x25 rectangle, insert 120 unit squares in it. Prove that one can place a circle with unit diameter in the rectangle, such that it doesn't intersect any of the squares.

This is a "there exists" problem. Therefore, I want to say that this problem is pigeonhole principle or probabilistic method. Hmmmm probabilistic method seems a little difficult... But I have no idea how pigeonhole principle works here either. Unfortunately, I can't really try small numbers either since the numbers 20,25,120 show no real pattern.

So, in short I have no idea as to how I should approach this problem.~

Looks like the question is using or something like that.

If this were the case, I don't see why pigeonhole principle wouldn't work (the bound isn't very tight from what I can tell?). Additionally, probabilistic method isn't screaming to be used either. I was only able to find a solution via contradiction (I decided to try contradiction as a last resort) by looking at the area where the center of the circle can't be.

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math puns are the first sine of madness
-JDR

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 Post subject: Re: Circles and Squares in RectanglesPosted: Fri, 25 May 2012 05:41:25 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
rdj5933mile5math64 wrote:
outermeasure wrote:
rdj5933mile5math64 wrote:
Given a 20x25 rectangle, insert 120 unit squares in it. Prove that one can place a circle with unit diameter in the rectangle, such that it doesn't intersect any of the squares.

This is a "there exists" problem. Therefore, I want to say that this problem is pigeonhole principle or probabilistic method. Hmmmm probabilistic method seems a little difficult... But I have no idea how pigeonhole principle works here either. Unfortunately, I can't really try small numbers either since the numbers 20,25,120 show no real pattern.

So, in short I have no idea as to how I should approach this problem.~

Looks like the question is using or something like that.

If this were the case, I don't see why pigeonhole principle wouldn't work (the bound isn't very tight from what I can tell?). Additionally, probabilistic method isn't screaming to be used either. I was only able to find a solution via contradiction (I decided to try contradiction as a last resort) by looking at the area where the center of the circle can't be.

Indeed, the area gives you the answer immediately (no need for contradiction).

You can also use pigeonhole plus a bit more.

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 Post subject: Re: Circles and Squares in RectanglesPosted: Fri, 25 May 2012 16:58:17 UTC
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Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
outermeasure wrote:
Indeed, the area gives you the answer immediately (no need for contradiction).

Oops you're right, you don't need contradiction.

outermeasure wrote:
You can also use pigeonhole plus a bit more.

Hmmm I searched for a pigeonhole solution for awhile, but had no luck.

How do you know when to use pigeonhole vs. other methods? What I mean is that probabilistic method is more general than pigeonhole principle. So it would be easier to use probabilistic method all of the time instead of considering the pigeonhole principle. However, I am afraid to rely on one method because I don't want to be too dependent on one approach.

_________________
math puns are the first sine of madness
-JDR

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 Post subject: Re: Circles and Squares in RectanglesPosted: Sat, 26 May 2012 05:42:52 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6005
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
rdj5933mile5math64 wrote:
outermeasure wrote:
Indeed, the area gives you the answer immediately (no need for contradiction).

Oops you're right, you don't need contradiction.

outermeasure wrote:
You can also use pigeonhole plus a bit more.

Hmmm I searched for a pigeonhole solution for awhile, but had no luck.

How do you know when to use pigeonhole vs. other methods? What I mean is that probabilistic method is more general than pigeonhole principle. So it would be easier to use probabilistic method all of the time instead of considering the pigeonhole principle. However, I am afraid to rely on one method because I don't want to be too dependent on one approach.

Using pigeonhole:
Spoiler:
20x25 can be split into 120 2x2's and a 20x1. If any of the 2x2's does not contain a centre of square, you are done.

Note that you can move the 20x1. So the only case to consider is ...

Take suitable slices, and you are reduced to ...

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 Post subject: Re: Circles and Squares in RectanglesPosted: Sat, 26 May 2012 16:13:27 UTC
 Senior Member

Joined: Wed, 4 Apr 2012 03:51:40 UTC
Posts: 129
Location: Hockeytown aka Detroit
outermeasure wrote:
Using pigeonhole:
Spoiler:
20x25 can be split into 120 2x2's and a 20x1. If any of the 2x2's does not contain a centre of square, you are done.

Note that you can move the 20x1. So the only case to consider is ...

Take suitable slices, and you are reduced to ...

Ok I was able to get it. Amazing as usual.

Wow! Hmmm I'm thinking about posting an NP-hard problem just to see if you can get it!

Thanks again outermeasure!

_________________
math puns are the first sine of madness
-JDR

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