# Minimum cuts to make a rectangle into a square, allowing bending

The puzzle is as follows:

Mike has a thin sheet of cardboard which is 96 centimeters large by 24 centimeters wide and a guillotine whose maximum cut length is 80 cm. Assuming this guillotine can cut a maximum of two layers of such cardboard. What is the minimum cuts he can make in order to get from that sheet 36 squares whose sides are of 8 centimeters?

The alternatives given are as follows:

1. 3 cuts
2. 5 cuts
3. 4 cuts
4. 2 cuts

I'm not sure how to get the optimal number of cuts. Does a way to find the optimization exist? So far I'm getting more than 6 cuts. My approach is to cut the bigger chunk of the cardboard in two pieces of 48 cm with hopes of getting a 6 by 6 squares, but from then on I got stuck.

Does a general guideline on how to solve this kind of puzzle exist?

This seems to be a modification from the usual problem which does not allow bending of the material to be cut.

This seems to be an adaptation from a reprinted copy of the 1970s edition of Martin Gardner's Puzzle Carnival found in a collection of riddles in Logical challenges.

Initially I thought that an approach would use the least common factor but upon looking at this puzzle it seems that is not the case. Since I'm not good with visualization of cuts, it would help a lot if the answer would include some sort of drawing or sketch on where should those cuts be made.

It can be done in

4

Picture contributed by @JaapScherphuis:

1. Bend along the short side and cut to get three equal 32x24 bits.

2.-4. (repeat 3 times cycling the three bits obtained in 1.) observe that 24+24+32=80 and arrange the pieces along the blade as follows: wide to be cut in the middle, wide to be cut at 1/4 and 3/4 and tall to be cut at 1/3 and 2/3.

Could it be done in fewer?

An optimal step would cut a combined total length of 2x80 or 20 in units of small square side. Altogether we need to cut 2x96 or 24 and 11x24 or 33. i.e. 57 in total. In principle, this would fit into 3 cuts, but since in the very first step we have to waste more than 3 units this cannot be done in practice, so 4 is indeed the best we can do.

• Very nice! It's also fairly straightforward to show that it cannot be done in even fewer cuts just by looking at the total length that needs to be cut, and how much each guillotine cut would have to achieve to do it in fewer. Mar 3, 2021 at 8:45
• You read my mind, @JaapScherphuis! Or I read yours. Mar 3, 2021 at 8:47
• @Albert.Lang Gee I still don't see it. I don't know how can I get three pieces of 32x24 bits by bending along the short side?. The rest I don't understand. For this reason I requested that a drawing to be included because for me it is difficult to get the right interpretation from the words. I don't know what you mean by cutting at 1/4, 3/4 and 1/3 and 2/3. Perhaps can you include a drawing please?. I'm lost. Mar 3, 2021 at 9:01
• Here's a picture Feel free to edit it into your answer. Mar 3, 2021 at 9:13
• I'm sorry @ChrisSteinbeckBell but this is beyond my fairly unimpressive drawing skills. Re the 32x24 pieces. Try first to imagine how you would get them by simply making two cuts. And then think about how you would have to bend to align the two cuts. Mar 3, 2021 at 9:14