Define the function $ f$ of the two numbers written on the board as: $ f(a,b)=a-b$ $ (a,b) \mapsto (a,\frac{a+b}{2})$ with $ f(a,\frac{a+b}{2})=a- \frac{a+b}{2}=\frac{a-b}{2}$ $ (a,b) \mapsto (\frac{a+b}{2},b)$ with $ f(\frac{a+b}{2},b)=\frac{a+b}{2}-b=\frac{a-b}{2}$ Then $ 2^{10}=1024 < 2000$ and $ 2^{11} \ge 2000$ then max amount of operations are $ 10$ with only example for $ m=2000-1024=976$ Moreover it is not difficult to generalize to any $ n\neq 2000$ or to give the set of numbers for having the possibility of performing $ k$ operations with $ k<10$.