A twisty puzzle is in a fully-scrambled state if it can no longer be solved if any pair of adjacent pieces is glued together. A puzzle can be fully scrambled if it has at least one fully-scrambled state.
The concept of "fully scrambled" was discussed at the Twisty Puzzles Forum. Quarkboy provided the following more comprehensive definition: A puzzle is in a fully scrambled state when no pair of adjacent pieces is in the same way adjacent as in a solved state at all times during a sequence of moves that solves the puzzle, for ALL possible sequences of moves that solve the puzzle from this state. Quarkboy also noted that there are multiple perspectives on the concept of fully scrambled:
- Mechanically / Group Theoretically: The above definition.
- Puzzle-ness-y: Is it in a state such that it takes some thinking to solve it? For example, a puzzle with 2 axis and only 2 moves at any time means there is never any choice in your move, either go forward or backward in the state chain which is a loop. That could be a puzzle which can be "fully scrambled" in the sense of #1 (you can split apart all the pieces), but can you really call it "scrambled" when there is no logic required to return them to the solved position?
- Competition Fairness: Speed solving cubes fairly requires some notion of a sufficiently scrambled cube for fairness based on some measure of how difficult a state is to solve. This would seem to me to be highly puzzle dependent.