Probably a question on pigeonhole principle
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
asked Jan 6 at 11:43
Funny guyFunny guy
116
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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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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
asked Jan 6 at 11:43
Funny guyFunny guy
116
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
add a comment |
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
asked Jan 6 at 11:43
Funny guyFunny guy
116
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
combinatorics pigeonhole-principle
asked Jan 6 at 11:43
Funny guyFunny guy
116
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.
asked Jan 6 at 11:43
Funny guyFunny guy
116
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.
edited Jan 6 at 12:14
Funny guy
asked Jan 6 at 11:43
Funny guyFunny guy
116
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.
asked Jan 6 at 11:43
Funny guyFunny guy
116
asked Jan 6 at 11:43
Funny guyFunny guy
116
116
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.
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
add a comment |
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
add a comment |
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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