So today I had a square paper with McDonalds coupons on it from which you had to cut out one coupon to use it. That got me wondering, how could I cut this square paper with coupons in least number of cuts by folding the paper over such that I cut every coupon out.
So here are the rules I made.
Assume there is a $NxN$ square made out of $1x1$ squares. What is the least number of cuts you have to make to cut every $1x1$ square "out" of the bigger $NxN$ square. You can fold the square in any way you like that is physically possible in the real world.
For clarity, in this picture the square paper is $3x3$ or $N=3$ and you want to cut out the 9 little squares out. If I wasn't clear enough, please ask me to clarify more.
Certainly you can always do it in 2 cuts. First fold the coupons vertically like an accordion so that all of the vertical lines to be cut lie on top of each other. Then it only takes one vertical cut to cut all of the vertical lines. Do the same thing with the horizontal lines and make one horizontal cut.
So the real question is can we do it in one cut? We can! Make all of the horizontal and vertical accordion folds as above so that you have one horizontal and one vertical cut to make, as shown below in red. By making two diagonal folds, we can make those lines lie on top of each other and make them both with one cut.
Of course in reality the paper might get way too thick to make so many folds and cut through them, but at least in theory it is possible to cut out all of your coupons with a single cut. Note this doesn't even rely on your coupon page being $N \times N$. It could be $N \times M$ instead, and the coupons might have varying heights and widths, as long as all the cuts to be made are vertical and horizontal going across the entire page.