Five Points on a Sphere

[gavinksong]

Prove that given any five points on a sphere, three of them lie on a hemisphere.

[BMAD]

Consider the great circle through any two of the points. This partitions the sphere into two hemispheres. By the pigeonhole principle, 2 of the remaining 3 points must lie in one of the hemispheres. These two points, along with the original two points, lie in a closed semi-sphere

[bonanova]

Let's place points on the globe with the intention of having no more then two of them in the northern [southern] hemisphere.

Our first four points must be on the equator, and we place them optimally at longitudes of 0, 90, 180 and 270 degrees.

If any three of them are considered to be north or south, we have failed, so let's say they are split 2 and 2.

Our fifth point must be one of the poles.In either case it will be the third point in its hemisphere.

 

I've finessed here the question of whether the hemisphere in the OP is open or closed.

[k-man]

Isn't this the same as

 

placing 5 coins into 2 pockets guarantees that one pocket is going to have at least 3 coins? Seems like the basic pigeonhole principle applies

[bonanova]

@k-man possibly so. The proof depends on whether the hemisphere is open or closed.

 

There are three pigeon holes,  e.g. northern and southern (open) hemispheres and the equator.

In this case neither of our proofs are strictly correct.

 

Of any 5 points on a sphere, some 4 of them lie on a closed hemisphere

This is a more interesting result, and it does follow from PHP.

[k-man]

@bonanova,

 

I guess I interpreted the OP as only the points are given and we choose the equator.

 

then you're right - whether hemispheres are open or closed changes the problem.

 

and we are free to choose the location of the equator after having the points given to us, then we can always choose the equator in such a way that open/closed issue becomes irrelevant. In other words, we choose the equator that doesn't pass through any of the 5 points. Then, by pigeonhole principle, at least 3 of 5 points will be in the same hemisphere.

[gavinksong]

My apologies to everybody. I meant to say four points in a closed hemisphere. 

BMAD still got it.

[BMAD]

yay. I so rarely successfully answer these puzzles.