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# Folding and Cutting Paper

## Question

 If a piece of rectangular paper is folded once only and a single straight cut is made in the following ways, how many pieces of paper will there be after the cut paper is unfolded in place? (1) A cut perpendicular to the (last) fold (2) A cut parallel to the (last) fold (3) A diagonal cut (4) A diagonal cut in the opposite direction (5) Both (1) and (2) (6) Both (3) and (4) (7) Both (5) and (6) The answers for cuts (1) to (7) are 2, 3, 3, 3, 6, 7, and 14, respectively. It is very difficult (rather practically impossible) to fold a piece of paper more than 6 or 7 times.   But, suppose we have a sheet of paper of infinite size and of negligible thickness, what would the outcome be if such a chunk of paper is folded 21 times and then cut accordingly to the mentioned seven ways? Each new fold from the start of folding is at right angles to the previous one. The images below show the different cuts to be made. The single fold in the given example (or the final fold when multiple folds are involved) appears on the right-hand side (||) of each drawing.

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Looks like 21 is a bit stiff. Let's change it to 13.

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