The conundrum was to prove that given five points on a sphere you can always find a closed hemisphere that contains four of them. Thanks to all of you who sent in great solutions! The first person to send one in was Daniel Phelps whose solution is published below. This is his third correct conundrum! Congratulations, Daniel!
Take any two of the 5 points on the sphere and make a great circle containing them, then the three remaining points will either be all on one of the hemispheres created by the great circle or at least 2 will be on one side. As long as there are at least two points in one of those hemispheres besides the two on the great circle, there will be at least 4 on the hemisphere including those on the great circle.