The optimal sequence of operations is very simple.

The optimal sequence of operations is adding $$$k-1$$$ 1-s into the set each time, at the same time decreasing $$$n$$$ by $$$k-1$$$. This implies that the answer is $$$\lceil \frac{n-1}{k-1}\rceil$$$.

"Most sequences" can be transformed into $$$[1]$$$. Conditions for a sequence to be un-transformable is stringent.

Find several simple substrings that make the string transformable.

We list some simple conditions for a string to be transformable:

• If 111 exists somewhere (as a substring) in the string, the string is always transformable.
• If 11 appears at least twice in the string, the string is always transformable.
• If the string both begins and ends with 1, it is always transformable.
• If the string begins or ends with 1 and 11 exists in the string, it is always transformable.

These can be found by simulating the operation for short strings on paper.

Contrarily, if a string does not meet any of the four items, it is always not transformable. This can be proved using induction (as an exercise).