In this answer I will show that five steps is the minimum number required (from the answer by @JoãoBravo).
Suppose by contradiction that there are four. Then the sequence will be of the form
F i v e, _ _ _ _, _ _ _ _, _ _ _ _, F o u r.
If the first letter remains unchanged the whole way through, the sequence is
F i v e, F _ _ _, F _ _ _, F _ _ _, F o u r.
For this case, please see the answer by @TimPillards, as the previous part of my answer has been demonstrated to be incorrect.
Now suppose that the first letter is changed to a letter ¬ the whole way through. Then the sequence is
F i v e, ¬ _ _ _, ¬ _ _ _, ¬ _ _ _, F o u r.
However, changing from i v e to o u r requires three steps, which are not allowed in this arrangement as the first and fourth steps are wasted.
Thus the only combinations left are
- F i v e, F _ _ _, ¬ _ _ _, ¬ _ _ _, F o u r.
This is also impossible as there are not enough steps (one) to change from i v e to o u r.
- F i v e, ¬ _ _ _, F _ _ _, F _ _ _, F o u r.
Steps two and three are pointless/a waste of steps so this case can be easily discarded.
- F i v e, F _ _ _, F _ _ _, ¬ _ _ _, F o u r.
Yet again, steps three and four are useless for the same reason above.
- F i v e, ¬ _ _ _, ¬ _ _ _, F _ _ _, F o u r.
There are not enough steps to make the transition i v e to o u r as the third step is wasted in converting ¬ to F.
- F i v e, F _ _ _, ¬ _ _ _, F _ _ _, F o u r.
Again, not sufficient due to the same reasoning above.
- F i v e, ¬ _ _ _, F _ _ _, ¬ _ _ _, F o u r.
This is also useless.
