Circles and Squares in Rectangles

rdj5933mile5math64 · **Posted:** Thu, 24 May 2012 18:11:34 UTC

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.~

outermeasure · **Posted:** Thu, 24 May 2012 18:22:39 UTC

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 $120=\lfloor\frac{20}{2}\rfloor\times\lfloor\frac{25}{2}\rfloor$ or something like that.

rdj5933mile5math64 · **Posted:** Fri, 25 May 2012 05:01:17 UTC

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 $120=\lfloor\frac{20}{2}\rfloor\times\lfloor\frac{25}{2}\rfloor$ 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.

outermeasure · **Posted:** Fri, 25 May 2012 05:41:25 UTC

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 $120=\lfloor\frac{20}{2}\rfloor\times\lfloor\frac{25}{2}\rfloor$ 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.

rdj5933mile5math64 · **Posted:** Fri, 25 May 2012 16:58:17 UTC

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.

outermeasure · **Posted:** Sat, 26 May 2012 05:42:52 UTC

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:

rdj5933mile5math64 · **Posted:** Sat, 26 May 2012 16:13:27 UTC

outermeasure wrote:

Using pigeonhole:

Spoiler:

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!

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Circles and Squares in Rectangles

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