The game is defined as follows.
Starting with a positive integer $N$, you can repeatedly perform one of two operations on $N$:
- If $N=ab$, replace $N$ by $\max(a,b)$.
- Subtract $1$ from $N$.
Find the minimum number of operations required to reduce $N$ to $0$.
After working for few $N$'s, I got that for any composite $N$, we can reduce $N$ to zero in a minimal number of steps if we change $N=\max(a,b) $ , where $ (a,b) $ is the pair in which $N=ab$, and $b$ is as small as possible, and $a \leq b$. For any prime $N$, we can only subtract 1, and continue with $N-1$.
For instance, if $N=48$, the pair described above is $(6, 8)$, leading to the solution
$$ 48 \to 8 \to 4 \to 2 \to 1 \to 0 $$
which is 5 moves long.
My strategy seems to be invalid. Why is that?