there is guaranteed to always be two antipodes on the Earth's surface - two points on exact opposite sides of the world - that have the exact same temperature and air pressure
1 is that the temperature and air pressure on earth are Continuous - in other words, there's no point where the temperature immediately jumps from hot to cold without some degree of gradient between the two
of course, some places will have the same temp/pressure as others, but we're fine with that. it just means multiple points on the sphere map to the same point on the 2D grid
anyway if you play with the restrictions of a loop made by two squiggles rotated 180 degrees from each other, you'll quickly find that you can't have the origin be outside of the loop
the second half will always wrap around and contain the origin. it is always inside the loop, unless the loop passes through the origin directly - in which point, hey, we found the point where g(p) = (0,0) already, so we're done
in any continuous mapping of a 4D hypersphere into a 3D space, there will exist antipodal points on the hypersphere that map to the same point on the plane
they want to cut up the necklace such that Alice and Bob can both take segments of it in such a way that each of them has the same number of each type of jewel
so, the cuts that are in mid-jewel will always be in pairs of the same color, so you can just slide them over and get the same result without cutting a jewel
and because the borsuk-ulam theorem applies to higher dimensions, this is still true as we add more colors of jewels to the necklace - each new color is just another dimension of the sphere
x = root(1/6)
y = root(1/3)
z = root(1/2)