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Number of lines through an $ m times n$ grid of points. [closed]
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Consider a $m times n$ grid of evenly spaced points. How many unique lines pass through at least 2 of these points?
I've found solutions for some special cases, but not the general case. Perhaps you all could help me?
combinatorics
asked Jul 17 at 22:37
Ando Bando
1343
closed as off-topic by Alex Francisco, Xander Henderson, Isaac Browne, Trần Thúc Minh TrÃ, Parcly Taxel Jul 18 at 3:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Alex Francisco, Xander Henderson, Isaac Browne, Trần Thúc Minh TrÃÂ, Parcly Taxel
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Consider a $m times n$ grid of evenly spaced points. How many unique lines pass through at least 2 of these points?
I've found solutions for some special cases, but not the general case. Perhaps you all could help me?
combinatorics
asked Jul 17 at 22:37
Ando Bando
1343
closed as off-topic by Alex Francisco, Xander Henderson, Isaac Browne, Trần Thúc Minh TrÃ, Parcly Taxel Jul 18 at 3:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Alex Francisco, Xander Henderson, Isaac Browne, Trần Thúc Minh TrÃÂ, Parcly Taxel
$mn choose 2$ is an upper bound.
– Will Sherwood
Jul 17 at 22:45
I'd guess you might be able to get an approximation for $m,n$ both large, but not a closed form.
– Thomas Andrews
Jul 17 at 23:40
You also might be able to get a recursive definition.
– Thomas Andrews
Jul 17 at 23:40
Please describe some of those "special cases" you found solutions for. I'm a little more optimistic than @ThomasAndrews that a closed form (of sorts) can be given.
– hardmath
Jul 18 at 2:14
@hardmath Simple things, 2 by n, 3 by n, 4 by n.
– Ando Bando
Jul 18 at 2:15
 |Â
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up vote
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down vote
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Consider a $m times n$ grid of evenly spaced points. How many unique lines pass through at least 2 of these points?
I've found solutions for some special cases, but not the general case. Perhaps you all could help me?
combinatorics
asked Jul 17 at 22:37
Ando Bando
1343
Consider a $m times n$ grid of evenly spaced points. How many unique lines pass through at least 2 of these points?
I've found solutions for some special cases, but not the general case. Perhaps you all could help me?
combinatorics
asked Jul 17 at 22:37
Ando Bando
1343
asked Jul 17 at 22:37
Ando Bando
1343
asked Jul 17 at 22:37
Ando Bando
1343
asked Jul 17 at 22:37
Ando Bando
1343
1343
closed as off-topic by Alex Francisco, Xander Henderson, Isaac Browne, Trần Thúc Minh TrÃ, Parcly Taxel Jul 18 at 3:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Alex Francisco, Xander Henderson, Isaac Browne, Trần Thúc Minh TrÃÂ, Parcly Taxel
closed as off-topic by Alex Francisco, Xander Henderson, Isaac Browne, Trần Thúc Minh TrÃ, Parcly Taxel Jul 18 at 3:07
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Alex Francisco, Xander Henderson, Isaac Browne, Trần Thúc Minh TrÃÂ, Parcly Taxel
$mn choose 2$ is an upper bound.
– Will Sherwood
Jul 17 at 22:45
I'd guess you might be able to get an approximation for $m,n$ both large, but not a closed form.
– Thomas Andrews
Jul 17 at 23:40
You also might be able to get a recursive definition.
– Thomas Andrews
Jul 17 at 23:40
Please describe some of those "special cases" you found solutions for. I'm a little more optimistic than @ThomasAndrews that a closed form (of sorts) can be given.
– hardmath
Jul 18 at 2:14
@hardmath Simple things, 2 by n, 3 by n, 4 by n.
– Ando Bando
Jul 18 at 2:15
 |Â
show 2 more comments
$mn choose 2$ is an upper bound.
– Will Sherwood
Jul 17 at 22:45
I'd guess you might be able to get an approximation for $m,n$ both large, but not a closed form.
– Thomas Andrews
Jul 17 at 23:40
You also might be able to get a recursive definition.
– Thomas Andrews
Jul 17 at 23:40
Please describe some of those "special cases" you found solutions for. I'm a little more optimistic than @ThomasAndrews that a closed form (of sorts) can be given.
– hardmath
Jul 18 at 2:14
@hardmath Simple things, 2 by n, 3 by n, 4 by n.
– Ando Bando
Jul 18 at 2:15
$mn choose 2$ is an upper bound.
– Will Sherwood
Jul 17 at 22:45
$mn choose 2$ is an upper bound.
– Will Sherwood
Jul 17 at 22:45
I'd guess you might be able to get an approximation for $m,n$ both large, but not a closed form.
– Thomas Andrews
Jul 17 at 23:40
I'd guess you might be able to get an approximation for $m,n$ both large, but not a closed form.
– Thomas Andrews
Jul 17 at 23:40
You also might be able to get a recursive definition.
– Thomas Andrews
Jul 17 at 23:40
You also might be able to get a recursive definition.
– Thomas Andrews
Jul 17 at 23:40
Please describe some of those "special cases" you found solutions for. I'm a little more optimistic than @ThomasAndrews that a closed form (of sorts) can be given.
– hardmath
Jul 18 at 2:14
Please describe some of those "special cases" you found solutions for. I'm a little more optimistic than @ThomasAndrews that a closed form (of sorts) can be given.
– hardmath
Jul 18 at 2:14
@hardmath Simple things, 2 by n, 3 by n, 4 by n.
– Ando Bando
Jul 18 at 2:15
@hardmath Simple things, 2 by n, 3 by n, 4 by n.
– Ando Bando
Jul 18 at 2:15
 |Â
show 2 more comments
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$mn choose 2$ is an upper bound.
– Will Sherwood
Jul 17 at 22:45
I'd guess you might be able to get an approximation for $m,n$ both large, but not a closed form.
– Thomas Andrews
Jul 17 at 23:40
You also might be able to get a recursive definition.
– Thomas Andrews
Jul 17 at 23:40
Please describe some of those "special cases" you found solutions for. I'm a little more optimistic than @ThomasAndrews that a closed form (of sorts) can be given.
– hardmath
Jul 18 at 2:14
@hardmath Simple things, 2 by n, 3 by n, 4 by n.
– Ando Bando
Jul 18 at 2:15