Probably a question on pigeonhole principle












-2














Suppose there are $q$ points $a_1,a_2,dots,a_q$ inside the closed unit square. Let $t_k$ be the minimal distance from $a_k$ to $a_j$ for any $j$ other than $k$. $$text{Show that }sum_{k=1}^q t_k^2le4$$



I'm unsure how to solve it, I think I'll probably need to use the pigeonhole principle, but I don't even understand the question completely to be able to do anything or make an effort to solve it.
And maybe, I don't even have to use the pigeonhole principle, I can't even guess what I have to use for this so sorry if the question tag is off.



Could someone explain how I'm to solve this in layman's terms?










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  • Hint: Can you figure out what the optimal arrangement of the points within the square is? That is the arrangement that minimises the sum of $r_{k}^2$, and then if you can show that in this case the inequality holds, by the pigeonhole principle you have it for every case.
    – Adam Higgins
    Jan 6 at 11:48










  • Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
    – José Carlos Santos
    Jan 6 at 11:49










  • Adam Higgins, I'm not even sure what the question actually is. Like I can't visualise what it's asking for. Maybe a bit of help understanding that first would be awesome!
    – Funny guy
    Jan 6 at 11:53






  • 1




    @AdamHiggins No, if we want to prove the inequality by (somehow!) finding an extremal arrangement, we need to find the arrangement that maximizes the sum.
    – David C. Ullrich
    Jan 6 at 14:59










  • @DavidC.Ullrich of course, my bad!
    – Adam Higgins
    Jan 6 at 15:00
















-2














Suppose there are $q$ points $a_1,a_2,dots,a_q$ inside the closed unit square. Let $t_k$ be the minimal distance from $a_k$ to $a_j$ for any $j$ other than $k$. $$text{Show that }sum_{k=1}^q t_k^2le4$$



I'm unsure how to solve it, I think I'll probably need to use the pigeonhole principle, but I don't even understand the question completely to be able to do anything or make an effort to solve it.
And maybe, I don't even have to use the pigeonhole principle, I can't even guess what I have to use for this so sorry if the question tag is off.



Could someone explain how I'm to solve this in layman's terms?










share|cite|improve this question









New contributor




Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • Hint: Can you figure out what the optimal arrangement of the points within the square is? That is the arrangement that minimises the sum of $r_{k}^2$, and then if you can show that in this case the inequality holds, by the pigeonhole principle you have it for every case.
    – Adam Higgins
    Jan 6 at 11:48










  • Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
    – José Carlos Santos
    Jan 6 at 11:49










  • Adam Higgins, I'm not even sure what the question actually is. Like I can't visualise what it's asking for. Maybe a bit of help understanding that first would be awesome!
    – Funny guy
    Jan 6 at 11:53






  • 1




    @AdamHiggins No, if we want to prove the inequality by (somehow!) finding an extremal arrangement, we need to find the arrangement that maximizes the sum.
    – David C. Ullrich
    Jan 6 at 14:59










  • @DavidC.Ullrich of course, my bad!
    – Adam Higgins
    Jan 6 at 15:00














-2












-2








-2







Suppose there are $q$ points $a_1,a_2,dots,a_q$ inside the closed unit square. Let $t_k$ be the minimal distance from $a_k$ to $a_j$ for any $j$ other than $k$. $$text{Show that }sum_{k=1}^q t_k^2le4$$



I'm unsure how to solve it, I think I'll probably need to use the pigeonhole principle, but I don't even understand the question completely to be able to do anything or make an effort to solve it.
And maybe, I don't even have to use the pigeonhole principle, I can't even guess what I have to use for this so sorry if the question tag is off.



Could someone explain how I'm to solve this in layman's terms?










share|cite|improve this question









New contributor




Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











Suppose there are $q$ points $a_1,a_2,dots,a_q$ inside the closed unit square. Let $t_k$ be the minimal distance from $a_k$ to $a_j$ for any $j$ other than $k$. $$text{Show that }sum_{k=1}^q t_k^2le4$$



I'm unsure how to solve it, I think I'll probably need to use the pigeonhole principle, but I don't even understand the question completely to be able to do anything or make an effort to solve it.
And maybe, I don't even have to use the pigeonhole principle, I can't even guess what I have to use for this so sorry if the question tag is off.



Could someone explain how I'm to solve this in layman's terms?







combinatorics pigeonhole-principle






share|cite|improve this question









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Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









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Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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share|cite|improve this question




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edited Jan 6 at 12:14







Funny guy













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asked Jan 6 at 11:43









Funny guyFunny guy

116




116




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New contributor





Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • Hint: Can you figure out what the optimal arrangement of the points within the square is? That is the arrangement that minimises the sum of $r_{k}^2$, and then if you can show that in this case the inequality holds, by the pigeonhole principle you have it for every case.
    – Adam Higgins
    Jan 6 at 11:48










  • Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
    – José Carlos Santos
    Jan 6 at 11:49










  • Adam Higgins, I'm not even sure what the question actually is. Like I can't visualise what it's asking for. Maybe a bit of help understanding that first would be awesome!
    – Funny guy
    Jan 6 at 11:53






  • 1




    @AdamHiggins No, if we want to prove the inequality by (somehow!) finding an extremal arrangement, we need to find the arrangement that maximizes the sum.
    – David C. Ullrich
    Jan 6 at 14:59










  • @DavidC.Ullrich of course, my bad!
    – Adam Higgins
    Jan 6 at 15:00


















  • Hint: Can you figure out what the optimal arrangement of the points within the square is? That is the arrangement that minimises the sum of $r_{k}^2$, and then if you can show that in this case the inequality holds, by the pigeonhole principle you have it for every case.
    – Adam Higgins
    Jan 6 at 11:48










  • Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
    – José Carlos Santos
    Jan 6 at 11:49










  • Adam Higgins, I'm not even sure what the question actually is. Like I can't visualise what it's asking for. Maybe a bit of help understanding that first would be awesome!
    – Funny guy
    Jan 6 at 11:53






  • 1




    @AdamHiggins No, if we want to prove the inequality by (somehow!) finding an extremal arrangement, we need to find the arrangement that maximizes the sum.
    – David C. Ullrich
    Jan 6 at 14:59










  • @DavidC.Ullrich of course, my bad!
    – Adam Higgins
    Jan 6 at 15:00
















Hint: Can you figure out what the optimal arrangement of the points within the square is? That is the arrangement that minimises the sum of $r_{k}^2$, and then if you can show that in this case the inequality holds, by the pigeonhole principle you have it for every case.
– Adam Higgins
Jan 6 at 11:48




Hint: Can you figure out what the optimal arrangement of the points within the square is? That is the arrangement that minimises the sum of $r_{k}^2$, and then if you can show that in this case the inequality holds, by the pigeonhole principle you have it for every case.
– Adam Higgins
Jan 6 at 11:48












Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
Jan 6 at 11:49




Welcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
– José Carlos Santos
Jan 6 at 11:49












Adam Higgins, I'm not even sure what the question actually is. Like I can't visualise what it's asking for. Maybe a bit of help understanding that first would be awesome!
– Funny guy
Jan 6 at 11:53




Adam Higgins, I'm not even sure what the question actually is. Like I can't visualise what it's asking for. Maybe a bit of help understanding that first would be awesome!
– Funny guy
Jan 6 at 11:53




1




1




@AdamHiggins No, if we want to prove the inequality by (somehow!) finding an extremal arrangement, we need to find the arrangement that maximizes the sum.
– David C. Ullrich
Jan 6 at 14:59




@AdamHiggins No, if we want to prove the inequality by (somehow!) finding an extremal arrangement, we need to find the arrangement that maximizes the sum.
– David C. Ullrich
Jan 6 at 14:59












@DavidC.Ullrich of course, my bad!
– Adam Higgins
Jan 6 at 15:00




@DavidC.Ullrich of course, my bad!
– Adam Higgins
Jan 6 at 15:00










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