Clicking the button internally calls this.render(), but that's not what actually causes the rendering to happen (you can see this in action because the text created by {Math.random()} doesn't change). However, if I simply call this.setState() instead of this.render(), it works fine.
So I guess my question is: do React components need to have state in order to rerender? Is there a way to force the component to update on demand without changing the state?