Circle Problem From 3Blue1Brown （分圆问题）

Background\text{Background}Background

Last night, lots of students from primary school came to our class to study OI.\text{Last night, lots of students from primary school came to our class to study OI.}Last night, lots of students from primary school came to our class to study OI.

Mark next to me was asked by one of them, "Dude, are you copying codes?"\text{Mark next to me was asked by one of them, "Dude, are you copying codes?"}Mark next to me was asked by one of them, "Dude, are you copying codes?"

Mark was very angry that time cuz he’s just moving his code from IDE to blog.\text{Mark was very angry that time cuz he's just moving his code from IDE to blog.}Mark was very angry that time cuz he’s just moving his code from IDE to blog.

So he decided to let them take a HARD test (though he failed).\text{So he decided to let them take a HARD test (though he failed).}So he decided to let them take a HARD test (though he failed).

Finally that guy apologized and we started upgrading our test so that it could \text{Finally that guy apologized and we started upgrading our test so that it could }Finally that guy apologized and we started upgrading our test so that it could be harder. They’ll take this test this Sunday.\text{be harder. They'll take this test this Sunday.}be harder. They’ll take this test this Sunday.

Problem\text{Problem}Problem

Interger N and N points in a circle are given. Connect every pair of \text{Interger }N\text{ and }N\text{ points in a circle are given. Connect every pair of }Interger N and N points in a circle are given. Connect every pair of 

these points to a edge. There aren’t any 3 edges which shares one point.\text{these points to a edge. There aren't any 3 edges which shares one point.}these points to a edge. There aren’t any 3 edges which shares one point.

Please calculate how many pieces of the circle are cut by these edges.\text{Please calculate how many pieces of the circle are cut by these edges.}Please calculate how many pieces of the circle are cut by these edges.

Solution\text{Solution}Solution

Let’s consider some cases with smaller Ns.\text{Let's consider some cases with smaller }N\text{s.}Let’s consider some cases with smaller Ns.

Easy to get\text{Easy to get}Easy to get

NNN	ansansans
111	111
222	222
333	444
444	888
555	161616
.........	.........

Dude, ans=2N−1. Solved.\text{Dude, }ans=2^{N-1}.\text{ Solved.}Dude, ans=2N−1. Solved.



But actually it’s wrong.\text{But actually it's wrong.}But actually it’s wrong.

These formula \text{These formula }These formula JUST right when N∈{1,2,3,4,5,6,10}.\text{ right when }N\in\{1,2,3,4,5,6,10\}. right when N∈{1,2,3,4,5,6,10}.

Let’s do some simple problems first.\text{Let's do some simple problems first.}Let’s do some simple problems first.

I. Calculate how many edges are there in the circle;\text{I. Calculate how many edges are there in the circle;}I. Calculate how many edges are there in the circle;

∵Every 2 points make a edge, and there’re N points,\because\text{Every 2 points make a edge, and there're }N\text{ points,}∵Every 2 points make a edge, and there’re N points,

∴There’re CN2 edges in total.\therefore\text{There're }C_{N}^{2}\text{ edges in total.}∴There’re CN2​ edges in total.

II. Calculate how many points of intersection of these edges.\text{II. Calculate how many points of intersection of these edges.}II. Calculate how many points of intersection of these edges.

It maybe a little difficult, but I think it’s necessary for you guys to think about it.\text{It maybe a little difficult, but I think it's necessary for you guys to think about it.}It maybe a little difficult, but I think it’s necessary for you guys to think about it.

∵Every 2 edges make a point of intersection, every 2 points make a edge,\because\text{Every 2 edges make a point of intersection, every 2 points make a edge,}∵Every 2 edges make a point of intersection, every 2 points make a edge,

and there’re N points,\text{and there're }N\text{ points,}and there’re N points,

∴There’re \therefore\text{There're }∴There’re N×(N−1)×(N−2)×(N−3)N\times(N-1)\times(N-2)\times(N-3)N×(N−1)×(N−2)×(N−3)=CN4 points of intersection in total.=C_{N}^{4}\text{ points of intersection in total.}=CN4​ points of intersection in total.



Here goes our Euler’s formula in topology. Set a polyhedron which has \text{Here goes our Euler's formula in topology. Set a polyhedron which has }Here goes our Euler’s formula in topology. Set a polyhedron which has 

V vertexes ,F pieces of surface, and E edges. It always meetV\text{ vertexes ,}F\text{ pieces of surface, and }E\text{ edges. It always meet}V vertexes ,F pieces of surface, and E edges. It always meetV−E+F=2V-E+F=2V−E+F=2

∴F=E−V+2.\therefore F=E-V+2.∴F=E−V+2.

Therefore, the answer is CN2+CN+4+2−1=CN2+CN+4+1. Solved.\text{Therefore, the answer is }C_{N}^{2}+C_{N}+{4}+2-1=C_{N}^{2}+C_{N}+{4}+1.\text{ Solved.}Therefore, the answer is CN2​+CN​+4+2−1=CN2​+CN​+4+1. Solved.

The End\text{The End}The End

Video by 3Blue1Brown from bilibili\text{Video by 3Blue1Brown from bilibili}Video by 3Blue1Brown from bilibili

Reference material\text{Reference material}Reference material