Why the points can't maintain social distance?

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Anindya Biswas
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Why the points can't maintain social distance?

Unread post by Anindya Biswas » Mon Jan 18, 2021 10:28 pm

$20$ points are taken inside a regular hexagon of side length $1$. Prove that there exists at least two points whose distance is less than or equal to $\frac12$.

I have tried to break the hexagon into several parts and apply PHP, but this seems a little tricky...
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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Mehrab4226
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Re: Why the points can't maintain social distance?

Unread post by Mehrab4226 » Wed Jan 20, 2021 11:33 pm

If PHP is useless in this, is there any way we can at least start to make progress? Because if there were 25 points then PHP would have worked fine. 20 points make it really difficult.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

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Anindya Biswas
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Re: Why the points can't maintain social distance?

Unread post by Anindya Biswas » Wed Jan 20, 2021 11:48 pm

Mehrab4226 wrote:
Wed Jan 20, 2021 11:33 pm
If PHP is useless in this, is there any way we can at least start to make progress? Because if there were 25 points then PHP would have worked fine. 20 points make it really difficult.
I can divide the hexagon into 18 pieces nicely, but those pieces aren't small enough :|
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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Mehrab4226
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Re: Why the points can't maintain social distance?

Unread post by Mehrab4226 » Thu Jan 21, 2021 12:21 am

Anindya Biswas wrote:
Wed Jan 20, 2021 11:48 pm

I can divide the hexagon into 18 pieces nicely, but those pieces aren't small enough :|
SAME! :|
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Asif Hossain
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Re: Why the points can't maintain social distance?

Unread post by Asif Hossain » Tue Feb 09, 2021 8:53 pm

21 points also work easily...(Though it can be divided into several hexagons to meet the condition Hint: If the side length of regular hex is a then the biggest corner is 2a)
Hmm..Hammer...Treat everything as nail

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Anindya Biswas
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Re: Why the points can't maintain social distance?

Unread post by Anindya Biswas » Tue Feb 09, 2021 11:04 pm

Asif Hossain wrote:
Tue Feb 09, 2021 8:53 pm
21 points also work easily...(Though it can be divided into several hexagons to meet the condition Hint: If the side length of regular hex is a then the biggest corner is 2a)
Yeah I actually figured this 20 points problem out some days after this post, now kept that solution as a secret for others to explore.
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

Asif Hossain
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Re: Why the points can't maintain social distance?

Unread post by Asif Hossain » Wed Feb 10, 2021 9:39 pm

:mrgreen: Actually i asked you that question on facebook and i inherited that ingenious solution from you
Hmm..Hammer...Treat everything as nail

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Anindya Biswas
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Re: Why the points can't maintain social distance?

Unread post by Anindya Biswas » Thu Feb 11, 2021 12:25 am

Asif Hossain wrote:
Wed Feb 10, 2021 9:39 pm
:mrgreen: Actually i asked you that question on facebook and i inherited that ingenious solution from you
Wow :o
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
John von Neumann

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