Numbers $1$ to $22$ are written on a board. A "move" is a procedure of picking two numbers $a,b$ on the board such that $b \geq a+2$, then erasing $a$ and $b$ to be replaced with $a+1$ and $b-1$. Determine the maximum possible number of moves that can be done on the board.
The maximum possible number of moves is $440$.
First we denote a move on $a,b$ as $\sigma(a,b)$.Define $S=\frac{1}{2}\sum_{i ~\textit{is on the board}}i^2$(calculate multiplicity).Notice that when we apply a move we decrease $S$ by at least $1$.We know that the sum of the numbers on the board is static,then by using the properties of convex function $f(x)=x^2$,we can get that $S$ achieve its minimum when there are $11~$ $11$s and $11~$ $12$s on the board.Then we can obtain the fact that the number of the moves is no more than $\left(\frac{1}{2} \sum_{i=1}^{22}i^2\right )-11\times (11^2+12^2)=440$.
On the other hand,the construction of $440$ moves is easy by induction.