(5 minutes or less)
First try the experiment with just the stick itself. Then tape the clay or weight somewhere on the stick and try again.
(5 minutes or more)
Support the stick by resting each of its ends on a finger. Slowly slide your fingers together until they meet. Your fingers will meet under the stick's center of gravity. Attach the weight or a piece of clay to some point on the stick. Again support the stick on two fingers, and then slide your fingers together to locate the new center of gravity. Move the weight or piece of clay to some new place on the stick. Repeat the experiment. Your fingers will always meet right under the center of gravity.
The stick's center of gravity is the place where you could balance the stick on just one finger. When you first support the stick with two fingers, in general one finger (the one that is closer to the center of gravity) will be holding a little more of the weight than the other. When you try to move your fingers closer together, the one that is carrying less weight will slide more easily. This finger will continue to slide more easily until it gets closer to the center of gravity than the other finger, at which point the situation will reverse and the other finger will begin to slide faster. Your left and right fingers simply alternate moving until they meet at the center of gravity, where both fingers support equal weight.