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Lattice triangles of positive area [closed]
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There are $n$ triangles of positive area that have one vertex at $(0,0)$ and their other two vertices with coordinates in $0,1,2,3,4$. Find the value of $n$.
I know that other than $(0,0)$ the vertices are
$$(0,1),(0,2),(0,3),(0,4),(1,0),(1,1),(1,2),(1,3),(1,4),(2,0),(2,1),(2,2),(2,3),(2,4),(3,0),(3,1),(3,2),(3,3),(3,4),(4,0),(4,1),(4,2),(4,3),(4,4)$$
but how do I select the two points which satisfy the given condition.
combinatorics
edited Jul 18 at 9:58
Parcly Taxel
33.6k136588
asked Jul 18 at 8:58
learner_avid
682413
closed as off-topic by Alex Francisco, Isaac Browne, Strants, Xander Henderson, José Carlos Santos Jul 19 at 22:54
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." РAlex Francisco, Isaac Browne, Strants, Xander Henderson, Jos̩ Carlos Santos
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up vote
2
down vote
favorite
There are $n$ triangles of positive area that have one vertex at $(0,0)$ and their other two vertices with coordinates in $0,1,2,3,4$. Find the value of $n$.
I know that other than $(0,0)$ the vertices are
$$(0,1),(0,2),(0,3),(0,4),(1,0),(1,1),(1,2),(1,3),(1,4),(2,0),(2,1),(2,2),(2,3),(2,4),(3,0),(3,1),(3,2),(3,3),(3,4),(4,0),(4,1),(4,2),(4,3),(4,4)$$
but how do I select the two points which satisfy the given condition.
combinatorics
edited Jul 18 at 9:58
Parcly Taxel
33.6k136588
asked Jul 18 at 8:58
learner_avid
682413
closed as off-topic by Alex Francisco, Isaac Browne, Strants, Xander Henderson, José Carlos Santos Jul 19 at 22:54
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." РAlex Francisco, Isaac Browne, Strants, Xander Henderson, Jos̩ Carlos Santos
Without restricting to say nonoverlapping triangles it seems no limit on $n.$ Also if they're drawn "at random" how can that determine $n$?
– coffeemath
Jul 18 at 9:09
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
There are $n$ triangles of positive area that have one vertex at $(0,0)$ and their other two vertices with coordinates in $0,1,2,3,4$. Find the value of $n$.
I know that other than $(0,0)$ the vertices are
$$(0,1),(0,2),(0,3),(0,4),(1,0),(1,1),(1,2),(1,3),(1,4),(2,0),(2,1),(2,2),(2,3),(2,4),(3,0),(3,1),(3,2),(3,3),(3,4),(4,0),(4,1),(4,2),(4,3),(4,4)$$
but how do I select the two points which satisfy the given condition.
combinatorics
edited Jul 18 at 9:58
Parcly Taxel
33.6k136588
asked Jul 18 at 8:58
learner_avid
682413
There are $n$ triangles of positive area that have one vertex at $(0,0)$ and their other two vertices with coordinates in $0,1,2,3,4$. Find the value of $n$.
I know that other than $(0,0)$ the vertices are
$$(0,1),(0,2),(0,3),(0,4),(1,0),(1,1),(1,2),(1,3),(1,4),(2,0),(2,1),(2,2),(2,3),(2,4),(3,0),(3,1),(3,2),(3,3),(3,4),(4,0),(4,1),(4,2),(4,3),(4,4)$$
but how do I select the two points which satisfy the given condition.
combinatorics
edited Jul 18 at 9:58
Parcly Taxel
33.6k136588
asked Jul 18 at 8:58
learner_avid
682413
edited Jul 18 at 9:58
Parcly Taxel
33.6k136588
edited Jul 18 at 9:58
Parcly Taxel
33.6k136588
edited Jul 18 at 9:58
Parcly Taxel
33.6k136588
33.6k136588
asked Jul 18 at 8:58
learner_avid
682413
asked Jul 18 at 8:58
learner_avid
682413
asked Jul 18 at 8:58
learner_avid
682413
682413
closed as off-topic by Alex Francisco, Isaac Browne, Strants, Xander Henderson, José Carlos Santos Jul 19 at 22:54
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." РAlex Francisco, Isaac Browne, Strants, Xander Henderson, Jos̩ Carlos Santos
closed as off-topic by Alex Francisco, Isaac Browne, Strants, Xander Henderson, José Carlos Santos Jul 19 at 22:54
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." РAlex Francisco, Isaac Browne, Strants, Xander Henderson, Jos̩ Carlos Santos
Without restricting to say nonoverlapping triangles it seems no limit on $n.$ Also if they're drawn "at random" how can that determine $n$?
– coffeemath
Jul 18 at 9:09
add a comment |Â
Without restricting to say nonoverlapping triangles it seems no limit on $n.$ Also if they're drawn "at random" how can that determine $n$?
– coffeemath
Jul 18 at 9:09
Without restricting to say nonoverlapping triangles it seems no limit on $n.$ Also if they're drawn "at random" how can that determine $n$?
– coffeemath
Jul 18 at 9:09
Without restricting to say nonoverlapping triangles it seems no limit on $n.$ Also if they're drawn "at random" how can that determine $n$?
– coffeemath
Jul 18 at 9:09
add a comment |Â
1 Answer
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The necessary and sufficient condition for the triangle to have positive area is to have the other two (non-origin) vertices not lie on the same line through the origin. That is, those vertices cannot be a subset of any of the following sets:
$$(0,1),(0,2),(0,3),(0,4)qquad(x=0)$$
$$(1,0),(2,0),(3,0),(4,0)qquad(y=0)$$
$$(1,1),(2,2),(3,3),(4,4)qquad(x=y)$$
$$(1,2),(2,4)qquad(2x=y)$$
$$(2,1),(4,2)qquad(2y=x)$$
There are $binom242=276$ possible lattice triangles, and those that have zero area number $3binom42+2binom22=20$. Therefore $n=276-20=256$.
answered Jul 18 at 10:06
Parcly Taxel
33.6k136588
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
The necessary and sufficient condition for the triangle to have positive area is to have the other two (non-origin) vertices not lie on the same line through the origin. That is, those vertices cannot be a subset of any of the following sets:
$$(0,1),(0,2),(0,3),(0,4)qquad(x=0)$$
$$(1,0),(2,0),(3,0),(4,0)qquad(y=0)$$
$$(1,1),(2,2),(3,3),(4,4)qquad(x=y)$$
$$(1,2),(2,4)qquad(2x=y)$$
$$(2,1),(4,2)qquad(2y=x)$$
There are $binom242=276$ possible lattice triangles, and those that have zero area number $3binom42+2binom22=20$. Therefore $n=276-20=256$.
answered Jul 18 at 10:06
Parcly Taxel
33.6k136588
add a comment |Â
up vote
2
down vote
The necessary and sufficient condition for the triangle to have positive area is to have the other two (non-origin) vertices not lie on the same line through the origin. That is, those vertices cannot be a subset of any of the following sets:
$$(0,1),(0,2),(0,3),(0,4)qquad(x=0)$$
$$(1,0),(2,0),(3,0),(4,0)qquad(y=0)$$
$$(1,1),(2,2),(3,3),(4,4)qquad(x=y)$$
$$(1,2),(2,4)qquad(2x=y)$$
$$(2,1),(4,2)qquad(2y=x)$$
There are $binom242=276$ possible lattice triangles, and those that have zero area number $3binom42+2binom22=20$. Therefore $n=276-20=256$.
answered Jul 18 at 10:06
Parcly Taxel
33.6k136588
add a comment |Â
up vote
2
down vote
up vote
2
down vote
The necessary and sufficient condition for the triangle to have positive area is to have the other two (non-origin) vertices not lie on the same line through the origin. That is, those vertices cannot be a subset of any of the following sets:
$$(0,1),(0,2),(0,3),(0,4)qquad(x=0)$$
$$(1,0),(2,0),(3,0),(4,0)qquad(y=0)$$
$$(1,1),(2,2),(3,3),(4,4)qquad(x=y)$$
$$(1,2),(2,4)qquad(2x=y)$$
$$(2,1),(4,2)qquad(2y=x)$$
There are $binom242=276$ possible lattice triangles, and those that have zero area number $3binom42+2binom22=20$. Therefore $n=276-20=256$.
answered Jul 18 at 10:06
Parcly Taxel
33.6k136588
The necessary and sufficient condition for the triangle to have positive area is to have the other two (non-origin) vertices not lie on the same line through the origin. That is, those vertices cannot be a subset of any of the following sets:
$$(0,1),(0,2),(0,3),(0,4)qquad(x=0)$$
$$(1,0),(2,0),(3,0),(4,0)qquad(y=0)$$
$$(1,1),(2,2),(3,3),(4,4)qquad(x=y)$$
$$(1,2),(2,4)qquad(2x=y)$$
$$(2,1),(4,2)qquad(2y=x)$$
There are $binom242=276$ possible lattice triangles, and those that have zero area number $3binom42+2binom22=20$. Therefore $n=276-20=256$.
answered Jul 18 at 10:06
Parcly Taxel
33.6k136588
answered Jul 18 at 10:06
Parcly Taxel
33.6k136588
answered Jul 18 at 10:06
Parcly Taxel
33.6k136588
answered Jul 18 at 10:06
Parcly Taxel
33.6k136588
33.6k136588
add a comment |Â
add a comment |Â
Without restricting to say nonoverlapping triangles it seems no limit on $n.$ Also if they're drawn "at random" how can that determine $n$?
– coffeemath
Jul 18 at 9:09