recurrence relation with min-max While solving a puzzle I got stuck with this recurrence 
\begin{align*}  &A(n) = \min_x \left \{ \max \left [ x,  1+A(n-x)
\right ] \right \} \\ &\text{where }A(1) = 1,A(0) = 0 \text{ and } 1 \leq x
\leq n , \text{ x is natural }\\ \end{align*}
Please give some idea how to proceed ?
Thanks !

Hint
Here are a few initial steps:


*

*Show by induction that $A(n) \le n$

*Show that when $n>1$, $\displaystyle\min_{1\le x\le n} = \min_{1\le x < n}$

*Use this to prove by induction that $A(n)\le A(n+1)$

*Prove that $A(n)\le A(n+1)\le 1 + A(n)$, so that $A(n+1) = A(n) \text{ or } 1+A(n)$

