How Many Points between two points?
$begingroup$
Given two points $A$ and $B$ on the $X-Y$-plane, I have to output the number of the lattice points on the segment $AB$. Note that $A$ and $B$ are also lattice point. Those who are confused with the definition of lattice point, lattice points are those points which have both $x$ and $y$ co-ordinate as an integer.
For example, for $A (3, 3)$ and $B (-1, -1)$ the output is $5$. The points are: $(-1, -1), (0, 0), (1, 1), (2, 2), (3, 3)$.
What is the procedure to solve this problem ?
number-theory
$endgroup$
add a comment |
$begingroup$
Given two points $A$ and $B$ on the $X-Y$-plane, I have to output the number of the lattice points on the segment $AB$. Note that $A$ and $B$ are also lattice point. Those who are confused with the definition of lattice point, lattice points are those points which have both $x$ and $y$ co-ordinate as an integer.
For example, for $A (3, 3)$ and $B (-1, -1)$ the output is $5$. The points are: $(-1, -1), (0, 0), (1, 1), (2, 2), (3, 3)$.
What is the procedure to solve this problem ?
number-theory
$endgroup$
add a comment |
$begingroup$
Given two points $A$ and $B$ on the $X-Y$-plane, I have to output the number of the lattice points on the segment $AB$. Note that $A$ and $B$ are also lattice point. Those who are confused with the definition of lattice point, lattice points are those points which have both $x$ and $y$ co-ordinate as an integer.
For example, for $A (3, 3)$ and $B (-1, -1)$ the output is $5$. The points are: $(-1, -1), (0, 0), (1, 1), (2, 2), (3, 3)$.
What is the procedure to solve this problem ?
number-theory
$endgroup$
Given two points $A$ and $B$ on the $X-Y$-plane, I have to output the number of the lattice points on the segment $AB$. Note that $A$ and $B$ are also lattice point. Those who are confused with the definition of lattice point, lattice points are those points which have both $x$ and $y$ co-ordinate as an integer.
For example, for $A (3, 3)$ and $B (-1, -1)$ the output is $5$. The points are: $(-1, -1), (0, 0), (1, 1), (2, 2), (3, 3)$.
What is the procedure to solve this problem ?
number-theory
number-theory
edited Jan 30 '15 at 18:08
Daniel W. Farlow
17.7k114588
17.7k114588
asked Feb 13 '13 at 5:02
Way to infinityWay to infinity
1,07711128
1,07711128
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
HINT: Let one of the points be $langle a,brangle$ and the other $langle c,drangle$; then the number of lattice points on the line segment joining them is the same as the number on the line segment joining $langle 0,0rangle$ to $langle c-a,d-brangle$. Thus, you might as well focus on counting the number of lattice points on the segment joining the origin to $langle m,nrangle$ for integers $m$ and $n$. Look at the equation of the line containing this segment: it’s
$$y=frac{n}mx;.$$
Suppose that when you reduce $frac{n}m$ to lowest terms, you get $frac{q}r$. Then your equation is
$$y=frac{q}rx;,$$
and $y$ is an integer if and only if $rmid x$.
Added: Suppose that the points are $langle -2,55rangle$ and $langle 1011,1055rangle$. I’d look at the segment from the origin to $langle 1011-(-2),1055-55rangle=langle 1013,1000rangle$. It lies on the line
$$y=frac{1000}{1013}x;.$$
Any lattice point on that line must have both $x$ and $y$ integers, so suppose that $x$ is an integer. When is $frac{1000}{1013}x$ an integer? The fraction is in lowest terms, so this occurs only when $x$ is a multiple of $1013$. On the other hand, we’re looking only at the segment between $langle 0,0$ and $langle 1013,1000rangle$, so clearly we must have $0le xle 1013$. How many multiples of $1013$ are there in this range? Just two, $0$ and $1013$. Thus, the endpoints are the only lattice points on that segment. Translating it parallel to itself up and to the left by adding $langle 2,-55rangle$ to restore the original endpoints doesn’t change the number of lattice points, so $langle -2,55rangle$ and $langle 1011,1055rangle$ are the only lattice points on the original segment.
$endgroup$
$begingroup$
Can you explain your answer for (a,b) = (-2,55) and (c,d)=(1011,1055) ?
$endgroup$
– Way to infinity
Feb 13 '13 at 5:22
$begingroup$
@SultanAhmedSagor: I’ve added a worked example based on those two points; it may help you to see what’s needed in the general case.
$endgroup$
– Brian M. Scott
Feb 13 '13 at 5:34
add a comment |
$begingroup$
Let me explain this with an example:
The line segment with endpoints (−9, −2) and (6, 8) has
slope [8 − (−2)]/[6 − (−9)] = 10/15 = 2/3
This means that starting at (−9, −2) and moving “up 2
and right 3” (corresponding to the rise and run of 2 and
3) repeatedly will give other points on the line that have
coordinates which are both integers.
These points are (−9, −2),(−6, 0),(−3, 2),(0, 4),(3, 6),(6, 8).
So far, this gives 6 points on the line with integer coordinates.
Are there any other such points?
If there were such a point between (−9, −2) and (6, 8), its y-coordinate would have to be equal to one of −1, 1, 3, 5, 7, the other integer possibilities between −2 and 8.
Consider the point on this line segment with y-coordinate 7.
Since this point has y-coordinate halfway between 6 and 8, then this point must be the midpoint of (3, 6) and (6, 8), which means that its x-coordinate is 1/2(3 + 6) = 4.5, which is not an integer.
In a similar way, the points on the line segment with y-coordinates −1, 1, 3, 5 do not have integer x-coordinates.
Therefore, the 6 points listed before are the only points on this line segment with integer coordinates.
source: https://www.cemc.uwaterloo.ca/contests/past_contests/2018/2018CayleySolution.pdf
New contributor
$endgroup$
$begingroup$
Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
yesterday
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f301890%2fhow-many-points-between-two-points%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
HINT: Let one of the points be $langle a,brangle$ and the other $langle c,drangle$; then the number of lattice points on the line segment joining them is the same as the number on the line segment joining $langle 0,0rangle$ to $langle c-a,d-brangle$. Thus, you might as well focus on counting the number of lattice points on the segment joining the origin to $langle m,nrangle$ for integers $m$ and $n$. Look at the equation of the line containing this segment: it’s
$$y=frac{n}mx;.$$
Suppose that when you reduce $frac{n}m$ to lowest terms, you get $frac{q}r$. Then your equation is
$$y=frac{q}rx;,$$
and $y$ is an integer if and only if $rmid x$.
Added: Suppose that the points are $langle -2,55rangle$ and $langle 1011,1055rangle$. I’d look at the segment from the origin to $langle 1011-(-2),1055-55rangle=langle 1013,1000rangle$. It lies on the line
$$y=frac{1000}{1013}x;.$$
Any lattice point on that line must have both $x$ and $y$ integers, so suppose that $x$ is an integer. When is $frac{1000}{1013}x$ an integer? The fraction is in lowest terms, so this occurs only when $x$ is a multiple of $1013$. On the other hand, we’re looking only at the segment between $langle 0,0$ and $langle 1013,1000rangle$, so clearly we must have $0le xle 1013$. How many multiples of $1013$ are there in this range? Just two, $0$ and $1013$. Thus, the endpoints are the only lattice points on that segment. Translating it parallel to itself up and to the left by adding $langle 2,-55rangle$ to restore the original endpoints doesn’t change the number of lattice points, so $langle -2,55rangle$ and $langle 1011,1055rangle$ are the only lattice points on the original segment.
$endgroup$
$begingroup$
Can you explain your answer for (a,b) = (-2,55) and (c,d)=(1011,1055) ?
$endgroup$
– Way to infinity
Feb 13 '13 at 5:22
$begingroup$
@SultanAhmedSagor: I’ve added a worked example based on those two points; it may help you to see what’s needed in the general case.
$endgroup$
– Brian M. Scott
Feb 13 '13 at 5:34
add a comment |
$begingroup$
HINT: Let one of the points be $langle a,brangle$ and the other $langle c,drangle$; then the number of lattice points on the line segment joining them is the same as the number on the line segment joining $langle 0,0rangle$ to $langle c-a,d-brangle$. Thus, you might as well focus on counting the number of lattice points on the segment joining the origin to $langle m,nrangle$ for integers $m$ and $n$. Look at the equation of the line containing this segment: it’s
$$y=frac{n}mx;.$$
Suppose that when you reduce $frac{n}m$ to lowest terms, you get $frac{q}r$. Then your equation is
$$y=frac{q}rx;,$$
and $y$ is an integer if and only if $rmid x$.
Added: Suppose that the points are $langle -2,55rangle$ and $langle 1011,1055rangle$. I’d look at the segment from the origin to $langle 1011-(-2),1055-55rangle=langle 1013,1000rangle$. It lies on the line
$$y=frac{1000}{1013}x;.$$
Any lattice point on that line must have both $x$ and $y$ integers, so suppose that $x$ is an integer. When is $frac{1000}{1013}x$ an integer? The fraction is in lowest terms, so this occurs only when $x$ is a multiple of $1013$. On the other hand, we’re looking only at the segment between $langle 0,0$ and $langle 1013,1000rangle$, so clearly we must have $0le xle 1013$. How many multiples of $1013$ are there in this range? Just two, $0$ and $1013$. Thus, the endpoints are the only lattice points on that segment. Translating it parallel to itself up and to the left by adding $langle 2,-55rangle$ to restore the original endpoints doesn’t change the number of lattice points, so $langle -2,55rangle$ and $langle 1011,1055rangle$ are the only lattice points on the original segment.
$endgroup$
$begingroup$
Can you explain your answer for (a,b) = (-2,55) and (c,d)=(1011,1055) ?
$endgroup$
– Way to infinity
Feb 13 '13 at 5:22
$begingroup$
@SultanAhmedSagor: I’ve added a worked example based on those two points; it may help you to see what’s needed in the general case.
$endgroup$
– Brian M. Scott
Feb 13 '13 at 5:34
add a comment |
$begingroup$
HINT: Let one of the points be $langle a,brangle$ and the other $langle c,drangle$; then the number of lattice points on the line segment joining them is the same as the number on the line segment joining $langle 0,0rangle$ to $langle c-a,d-brangle$. Thus, you might as well focus on counting the number of lattice points on the segment joining the origin to $langle m,nrangle$ for integers $m$ and $n$. Look at the equation of the line containing this segment: it’s
$$y=frac{n}mx;.$$
Suppose that when you reduce $frac{n}m$ to lowest terms, you get $frac{q}r$. Then your equation is
$$y=frac{q}rx;,$$
and $y$ is an integer if and only if $rmid x$.
Added: Suppose that the points are $langle -2,55rangle$ and $langle 1011,1055rangle$. I’d look at the segment from the origin to $langle 1011-(-2),1055-55rangle=langle 1013,1000rangle$. It lies on the line
$$y=frac{1000}{1013}x;.$$
Any lattice point on that line must have both $x$ and $y$ integers, so suppose that $x$ is an integer. When is $frac{1000}{1013}x$ an integer? The fraction is in lowest terms, so this occurs only when $x$ is a multiple of $1013$. On the other hand, we’re looking only at the segment between $langle 0,0$ and $langle 1013,1000rangle$, so clearly we must have $0le xle 1013$. How many multiples of $1013$ are there in this range? Just two, $0$ and $1013$. Thus, the endpoints are the only lattice points on that segment. Translating it parallel to itself up and to the left by adding $langle 2,-55rangle$ to restore the original endpoints doesn’t change the number of lattice points, so $langle -2,55rangle$ and $langle 1011,1055rangle$ are the only lattice points on the original segment.
$endgroup$
HINT: Let one of the points be $langle a,brangle$ and the other $langle c,drangle$; then the number of lattice points on the line segment joining them is the same as the number on the line segment joining $langle 0,0rangle$ to $langle c-a,d-brangle$. Thus, you might as well focus on counting the number of lattice points on the segment joining the origin to $langle m,nrangle$ for integers $m$ and $n$. Look at the equation of the line containing this segment: it’s
$$y=frac{n}mx;.$$
Suppose that when you reduce $frac{n}m$ to lowest terms, you get $frac{q}r$. Then your equation is
$$y=frac{q}rx;,$$
and $y$ is an integer if and only if $rmid x$.
Added: Suppose that the points are $langle -2,55rangle$ and $langle 1011,1055rangle$. I’d look at the segment from the origin to $langle 1011-(-2),1055-55rangle=langle 1013,1000rangle$. It lies on the line
$$y=frac{1000}{1013}x;.$$
Any lattice point on that line must have both $x$ and $y$ integers, so suppose that $x$ is an integer. When is $frac{1000}{1013}x$ an integer? The fraction is in lowest terms, so this occurs only when $x$ is a multiple of $1013$. On the other hand, we’re looking only at the segment between $langle 0,0$ and $langle 1013,1000rangle$, so clearly we must have $0le xle 1013$. How many multiples of $1013$ are there in this range? Just two, $0$ and $1013$. Thus, the endpoints are the only lattice points on that segment. Translating it parallel to itself up and to the left by adding $langle 2,-55rangle$ to restore the original endpoints doesn’t change the number of lattice points, so $langle -2,55rangle$ and $langle 1011,1055rangle$ are the only lattice points on the original segment.
edited Oct 18 '15 at 8:46
Paul Tarjan
1034
1034
answered Feb 13 '13 at 5:07
Brian M. ScottBrian M. Scott
458k38512915
458k38512915
$begingroup$
Can you explain your answer for (a,b) = (-2,55) and (c,d)=(1011,1055) ?
$endgroup$
– Way to infinity
Feb 13 '13 at 5:22
$begingroup$
@SultanAhmedSagor: I’ve added a worked example based on those two points; it may help you to see what’s needed in the general case.
$endgroup$
– Brian M. Scott
Feb 13 '13 at 5:34
add a comment |
$begingroup$
Can you explain your answer for (a,b) = (-2,55) and (c,d)=(1011,1055) ?
$endgroup$
– Way to infinity
Feb 13 '13 at 5:22
$begingroup$
@SultanAhmedSagor: I’ve added a worked example based on those two points; it may help you to see what’s needed in the general case.
$endgroup$
– Brian M. Scott
Feb 13 '13 at 5:34
$begingroup$
Can you explain your answer for (a,b) = (-2,55) and (c,d)=(1011,1055) ?
$endgroup$
– Way to infinity
Feb 13 '13 at 5:22
$begingroup$
Can you explain your answer for (a,b) = (-2,55) and (c,d)=(1011,1055) ?
$endgroup$
– Way to infinity
Feb 13 '13 at 5:22
$begingroup$
@SultanAhmedSagor: I’ve added a worked example based on those two points; it may help you to see what’s needed in the general case.
$endgroup$
– Brian M. Scott
Feb 13 '13 at 5:34
$begingroup$
@SultanAhmedSagor: I’ve added a worked example based on those two points; it may help you to see what’s needed in the general case.
$endgroup$
– Brian M. Scott
Feb 13 '13 at 5:34
add a comment |
$begingroup$
Let me explain this with an example:
The line segment with endpoints (−9, −2) and (6, 8) has
slope [8 − (−2)]/[6 − (−9)] = 10/15 = 2/3
This means that starting at (−9, −2) and moving “up 2
and right 3” (corresponding to the rise and run of 2 and
3) repeatedly will give other points on the line that have
coordinates which are both integers.
These points are (−9, −2),(−6, 0),(−3, 2),(0, 4),(3, 6),(6, 8).
So far, this gives 6 points on the line with integer coordinates.
Are there any other such points?
If there were such a point between (−9, −2) and (6, 8), its y-coordinate would have to be equal to one of −1, 1, 3, 5, 7, the other integer possibilities between −2 and 8.
Consider the point on this line segment with y-coordinate 7.
Since this point has y-coordinate halfway between 6 and 8, then this point must be the midpoint of (3, 6) and (6, 8), which means that its x-coordinate is 1/2(3 + 6) = 4.5, which is not an integer.
In a similar way, the points on the line segment with y-coordinates −1, 1, 3, 5 do not have integer x-coordinates.
Therefore, the 6 points listed before are the only points on this line segment with integer coordinates.
source: https://www.cemc.uwaterloo.ca/contests/past_contests/2018/2018CayleySolution.pdf
New contributor
$endgroup$
$begingroup$
Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
yesterday
add a comment |
$begingroup$
Let me explain this with an example:
The line segment with endpoints (−9, −2) and (6, 8) has
slope [8 − (−2)]/[6 − (−9)] = 10/15 = 2/3
This means that starting at (−9, −2) and moving “up 2
and right 3” (corresponding to the rise and run of 2 and
3) repeatedly will give other points on the line that have
coordinates which are both integers.
These points are (−9, −2),(−6, 0),(−3, 2),(0, 4),(3, 6),(6, 8).
So far, this gives 6 points on the line with integer coordinates.
Are there any other such points?
If there were such a point between (−9, −2) and (6, 8), its y-coordinate would have to be equal to one of −1, 1, 3, 5, 7, the other integer possibilities between −2 and 8.
Consider the point on this line segment with y-coordinate 7.
Since this point has y-coordinate halfway between 6 and 8, then this point must be the midpoint of (3, 6) and (6, 8), which means that its x-coordinate is 1/2(3 + 6) = 4.5, which is not an integer.
In a similar way, the points on the line segment with y-coordinates −1, 1, 3, 5 do not have integer x-coordinates.
Therefore, the 6 points listed before are the only points on this line segment with integer coordinates.
source: https://www.cemc.uwaterloo.ca/contests/past_contests/2018/2018CayleySolution.pdf
New contributor
$endgroup$
$begingroup$
Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
yesterday
add a comment |
$begingroup$
Let me explain this with an example:
The line segment with endpoints (−9, −2) and (6, 8) has
slope [8 − (−2)]/[6 − (−9)] = 10/15 = 2/3
This means that starting at (−9, −2) and moving “up 2
and right 3” (corresponding to the rise and run of 2 and
3) repeatedly will give other points on the line that have
coordinates which are both integers.
These points are (−9, −2),(−6, 0),(−3, 2),(0, 4),(3, 6),(6, 8).
So far, this gives 6 points on the line with integer coordinates.
Are there any other such points?
If there were such a point between (−9, −2) and (6, 8), its y-coordinate would have to be equal to one of −1, 1, 3, 5, 7, the other integer possibilities between −2 and 8.
Consider the point on this line segment with y-coordinate 7.
Since this point has y-coordinate halfway between 6 and 8, then this point must be the midpoint of (3, 6) and (6, 8), which means that its x-coordinate is 1/2(3 + 6) = 4.5, which is not an integer.
In a similar way, the points on the line segment with y-coordinates −1, 1, 3, 5 do not have integer x-coordinates.
Therefore, the 6 points listed before are the only points on this line segment with integer coordinates.
source: https://www.cemc.uwaterloo.ca/contests/past_contests/2018/2018CayleySolution.pdf
New contributor
$endgroup$
Let me explain this with an example:
The line segment with endpoints (−9, −2) and (6, 8) has
slope [8 − (−2)]/[6 − (−9)] = 10/15 = 2/3
This means that starting at (−9, −2) and moving “up 2
and right 3” (corresponding to the rise and run of 2 and
3) repeatedly will give other points on the line that have
coordinates which are both integers.
These points are (−9, −2),(−6, 0),(−3, 2),(0, 4),(3, 6),(6, 8).
So far, this gives 6 points on the line with integer coordinates.
Are there any other such points?
If there were such a point between (−9, −2) and (6, 8), its y-coordinate would have to be equal to one of −1, 1, 3, 5, 7, the other integer possibilities between −2 and 8.
Consider the point on this line segment with y-coordinate 7.
Since this point has y-coordinate halfway between 6 and 8, then this point must be the midpoint of (3, 6) and (6, 8), which means that its x-coordinate is 1/2(3 + 6) = 4.5, which is not an integer.
In a similar way, the points on the line segment with y-coordinates −1, 1, 3, 5 do not have integer x-coordinates.
Therefore, the 6 points listed before are the only points on this line segment with integer coordinates.
source: https://www.cemc.uwaterloo.ca/contests/past_contests/2018/2018CayleySolution.pdf
New contributor
New contributor
answered yesterday
Arqam AdiyanArqam Adiyan
1
1
New contributor
New contributor
$begingroup$
Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
yesterday
add a comment |
$begingroup$
Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
yesterday
$begingroup$
Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
yesterday
$begingroup$
Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
$endgroup$
– GNUSupporter 8964民主女神 地下教會
yesterday
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f301890%2fhow-many-points-between-two-points%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown