# PSE Advent Calendar 2022 (Day 17) Priority polar problem

This puzzle is part of the Puzzling Stack Exchange Advent Calendar 2022. The accepted answer to this question will be awarded a bounty worth 50 reputation.

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Dear puzzlers

I have a big problem: A worldwide disaster is about to happen.
Our trash cans are almost full. And in addition, we are short on paper.
Since you all are so good at finding things out, can you please reduce our waste problem?

Some context:
The cause of the problem is that our paper recycling machine broke down. It is not repairable anymore.

Of course, I immediately ordered a new one, but intensive usage and the temperature situation forces me to order it from the specialists. The pinguins immediately went to work and are shipping a new cold resistant paper recycling machine as we speak, but as you can imagine delivery times are quite bad.

I have considered starting up the paper factory during wintertime, but as you probably know it is permanently night here during winter. The darkness makes it quite hard to find and process trees here at the NorthPole, and without wood we cannot make extra paper.

So, I see only one immediate solution: Be less wasteful while creating the paper wrappings of our gifts.

PS: Our working process:

• Each elf gets a rectangular piece of paper, cuts a wrapping out of it for a rectangular present; folds it around said present, and throws the leftover paper in the trashcan so it can be recycled
• The size of the present is quite flexible, but the longest edge should not be more than twice the length of the shortest edge.
• In this emergency, the elves should be capable of making 4 wrappings out of 1 piece of paper. More will be too complicated and just lead to accidents, we do not want that! The 4 wrappings do not have to be the same.

Αny reduction in waste will be very appreciated and immediately incorporated in our process. If the waste can be reduced to half quickly, Christmas should be safe.

I assume everything is clear now, but to avoid any confusion and or delay some examples below.

Addition: After rereading my question I want to be more precise about something: The working process says it correctly, the picture text in less accurate regarding: The 2 to 1 ratio refers to wrappings (only), e.g. combining wrappings for 1x1x2 presents with wrappings for 10x10x10 presents is allowed.

hint 1:

An opportunity not yet used in the current answers is that one may combine different sized wrappings. Also utilizing this allows an even better solution (than the currently existing 2.)

hint 2:

Perfectly fitting one side of a wrapping onto another one almost halves the waste, but most holes can be more than 3/4 filled with a smaller wrapping.

hint 3:

Nonrectangular quadrilaterals can often be filled well with a wrapping since the outer parts of a wrapping can be cut nonorthogonal (and the cut of part attached elsewhere.

for completeness: hint/suggestion 4

When placing a (small) wrapping 'inside' a larger one, tilting the smaller one in combination moving parts of the larger one is often optimal.

PS:

Note that a small rotation (like in some answers) even improves the best orthogonal solution where only 1 wrapping is cut from a rectangular area.

• Would it be cheating to wrap a present with 6 separate pieces of paper and a bit of tape? Dec 17, 2022 at 9:38
• Yes, "Each wrapping must be one piece" i.m.o. implies that. The idea is a no waste solution is not possible. I also added the 2-1 length restriction esp. to prevent 'perfect' solutions. Dec 17, 2022 at 9:44
• Ah, yes, I didn't see that sentence. Sorry. Dec 17, 2022 at 12:11
• Was "pinguins" intended as a pun on "penguins"? Dec 20, 2022 at 14:59
• No, that slipped through, I am Dutch Dec 20, 2022 at 15:14

A refined version of @loopywalt's answer taking advantage of @Retudin's recent clarifications. Waste is down to

approx. 0.457 %

Please be sure to enjoy the way in which the two large wraps not only complement each other where they are joined but do so creating just the right horizontal offset.

• Wow, far nicer and better than I had in mind. There is still room to improve though. It hardly matters here because it fits so well, but there is a small bit of maneuvering room for the small wrapping. Rotating them can reduce to 0.454% (maybe 0.453) Dec 21, 2022 at 9:46
• @Retudin thanks. I'm quite happy with it myself. (Especially with the Tetro/Pentomino cut.) What solution did you have in mind? You mention it was better than the two other answers? Dec 21, 2022 at 11:34
• I thought I needed the same length i.e. 2x1x1 like loopy's answer, and mixed 2 big with 2 small for a little over 2% waste due to the required angle not being perfect. Dec 21, 2022 at 12:00

A refined version of @Florian F's. answer cutting waste to just under

3%

For a start, you can do

just below 5% waste.

And this is how

All presents are sized 1x1x2. The waste is 1/21 or ~4.8%

• Well done. You saved Christmas. I will let the elves use this as long as no better solution is provided. (I feel a bit stupid since this solution was not intended to Save Christmas already, but well, the earlier Christmas is saved the better.) Dec 17, 2022 at 14:06