Berdonikov A.

Let's call a natural number good if all its digits are non-zero. A good number is called special if it has at least $k$ digits and the digits are in strictly increasing order (from left to right).

Suppose we have some good number. In one move, it is allowed to append a special number to either end or insert it between any two of its digits, or, conversely, erase a special number from its representation. For what largest $k$ can any good number be transformed into any other good number using such moves?
Obviously, a special number does not have more than nine digits. If $k=9$, then with each operation, the number of digits changes by exactly 9, meaning the remainder of the number of its digits divided by 9 does not change, and a two-digit number cannot be made from a one-digit number.

Let $k=8$. Since all operations are reversible, it is sufficient to prove that any digit can be inserted.

We will prove by induction on $n$ that we can insert the digit $n$.

Base case. To insert 1, first insert 123456789, and then erase 23456789.

Inductive step. Suppose we can insert (and erase) digits from 1 to $n-1<9$. To insert $n$, first insert 123456789, erase $12 \ldots(n-1)$ one by one, then insert $12 \ldots(n-1)$ one by one to the right of $n$, and finally erase the number $12 \ldots(n-1)(n+1) \ldots 9$ (for $n=9$, the last two actions are not needed, and for $n=8$, the number 12...79 is erased).

## Answer

For $k=8$.