Answer
The fingers take turns supporting less weight as they alternately slide closer to the CG, and eventually meet there.
Work Step by Step
Your fingers start out on either side of the CG. As your fingers move, one of them randomly supports more of the meterstick’s weight. The other finger exerts a smaller normal force on the stick, and a correspondingly smaller frictional force. The meterstick slides more easily and that finger moves closer to the center of gravity.
