Suppose to each vertex of a pentagon, we assign a number $\displaystyle x_i $ with $\displaystyle s = \sum x_i >0 $. If $\displaystyle x,y,z $ are numbers assigned to three successive vertices and if $\displaystyle y<0 $, then replace $\displaystyle (x,y,z) $ with $\displaystyle (x+y, -y, y+z) $. Repeat this as long as $\displaystyle y<0 $. Does this algorithm always stop? If it does stop, how many steps are needed? If not, why doesn't it stop? Prove that it stops or does not stop.