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Bob can satisfy any Alice's request with 5 pieces of paper: two rectangles 8x4 and 4x2, two squares 4x4 and 2x2 and one piece of three unit squares in an L shape.
I'm not sure, however, if that is an actual minimum.

Image:

EDIT
The answer by elias is much better than mine. Is is based on the same concept of iterative covering the un-punched area, but...

it covers 3/4 of a remaining figure in each step:

while I have stopped at covering just a half (except the last step, where I cover last 3 squares in a single L-piece).

Bob can satisfy any Alice's request with 5 pieces of paper: two rectangles 8x4 and 4x2, two squares 4x4 and 2x2 and one piece of three unit squares in an L shape.
I'm not sure, however, if that is an actual minimum.

Image:

EDIT
The answer by elias is much better than mine. Is is based on the same concept of iterative covering the un-punched area, but...

it covers 3/4 of a remaining figure in each step:

while I have stopped at covering just a half (except the last step, where I cover last 3 squares in a single L-piece).

Bob can satisfy any Alice's request with 5 pieces of paper: two rectangles 8x4 and 4x2, two squares 4x4 and 2x2 and one piece of three unit squares in an L shape.
I'm not sure, however, if that is an actual minimum.

Image:

Bob can satisfy any Alice's request with 5 pieces of paper: two rectangles 8x4 and 4x2, two squares 4x4 and 2x2 and one piece of three unit squares in an L shape.
I'm not sure, however, if that is an actual minimum.

Image:

EDIT
The answer by elias is much better than mine. Is is based on the same concept of iterative covering the un-punched area, but...

it covers 3/4 of a remaining figure in each step:

while I have stopped at covering just a half (except the last step, where I cover last 3 squares in a single L-piece).

Bob can satisfy any Alice's request with 5 pieces of paper.
I'm not sure, however, if that is an actual minimum.

Bob can satisfy any Alice's request with 5 pieces of paper: two rectangles 8x4 and 4x2, two squares 4x4 and 2x2 and one piece of three unit squares in an L shape.
I'm not sure, however, if that is an actual minimum.

Image: