You been wrapping Christmas presents haven't you?

I have indeed 

Original Sheet: |----|----|----| | 1  | 2  | 3  | |----|----|----| | 4  | 5  | 6  | |----|----|----| | 7  | 8  | 9  | |----|----|----| Cuts: |----|----|----| | 1  | 2  | 3  | |----|++++++ | 4  | 5  | 6  | |----|++++++ | 7  | 8  | 9  | |----|----|----|                                                 ...

With slight re-wording this would be a generalization of the Monte Hall problem.  E.g. There are N chairs with 1 prize randomly placed beneath. You pick a chair at random.  K chairs are removed--including neither your choice nor the winning chair. Q1: What is the probability your chair is the winner...