This is a dynamical system. The important idea here is that there are a finite number of states so the system must 
eventually reach a state it has been in before and subsequently the states must change in a periodic cycle. It may 
reach a steady (fixed) state or it may cycle through the same p states (where <i>p</i> &#x003E; 1) in the same order over and over 
again, a cycle of length <i>p</i> (a p-cycle). If the number of beads is a power of 2 then the system always reduces to a 
steady state with all red beads whatever the initial state but the proof is rather subtle.