Consider a tunnel (not necessarily straight) of length 101 units through a
sphere of radius 51 units.
Prove that the sphere can be cut into two congruent pieces one of which
contains the tunnel in its interior.
Let the entry and exit points be A and B. Then all posible paths of length
101 lie in an ellipsoid such that the sum of the distance from A and that from
B is less than 101 units. But the centre of the sphere cannot be inside this
ellipsoid as the sum of the distance from A to the centre and from B to the
centre is 102.
Now the plane through the centre normal to the plane containing the points
A, B and the centre does not intersect the ellipsoid, and so divides the sphere
(ignoring the hole) into two equal hemisphere.