# Gifts



## rokicki (Aug 24, 2010)

If you're not allowed to cut the paper, what's the smallest piece of wrapping paper (by area) that you can use to wrap a standard Rubik's cube that is 57mm on a side?


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## qqwref (Aug 24, 2010)

We're assuming it just has to cover the surface, right? I know how to do it with a square of side length 2 sqrt(2) * 57mm.


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## guzman (Aug 24, 2010)

qqwref said:


> We're assuming it just has to cover the surface, right? I know how to do it with a square of side length 2 sqrt(2) * 57mm.



I also found your solution and a different one that uses a rectangle
2*57mm x 4*57mm
and has therefore the same area as yours.
I've been trying for a while but I couldn't fing anything better.
I thought this video with very interesting cube unfoldings could be useful ... but I didn't reach any new solutions.

guzman.


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## Cride5 (Aug 25, 2010)

It's possible to completely cover the cube with a rectangular strip of paper with dimensions: 7*57mm x 57mm (area of 22743 mm²)



Spoiler



To do it, place the D-face over the centre of the paper strip, wrap each end round the F/B faces, so that they meet on the U-face. Where they meet on the U-face, make a 45° diagonal fold on each end, so that one end folds over towards the L-face, and the other folds over towards the R-face. 

5/6 of the faces are covered with just 1 layer of wrapping paper, and the U-face, is covered with 2 layers.


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## qqwref (Aug 25, 2010)

Mm, good point.


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## Lorken (Aug 25, 2010)

Make sure the person is interested in cubes first, I got my brother an Edison cube, and he gave it away to a friend!
To wrap it up though, its basically what Cride5 said.
A rectangle about 400mm x 220mm should do the trick though. (About that size anyway)


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## rokicki (Aug 25, 2010)

Very nice solutions so far!

I think there are some additional solutions not yet considered.

Cride5's solution is definitely new to me; I had not thought that was possible. I may use as a puzzle in the future.

I intentionally left the definition of the problem a little vague to see how people riff on it.


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## rokicki (Aug 25, 2010)

qqwref said:


> We're assuming it just has to cover the surface, right? I know how to do it with a square of side length 2 sqrt(2) * 57mm.



This solution I came up with quite by surprise about a month ago when trying to wrap a birthday gift for a nephew. My paper was not large enough to wrap it the normal rectangular way. I was very happy when I discovered this trick, and was able to wrap the gift completely and nicely without needing to go to the store and buy new wrapping paper.


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## Stefan (Aug 29, 2010)

Using a rectangle, I can get as close to the cube surface area as I want. For example, I can wrap a 10x10x10 using a strip one piece wide and 646 pieces long, that's less than half a side "wasted". In general, for an NxNxN, a strip one piece wide and 6*N*N + 5*N-4 pieces long suffices. For large N, this approaches the cube surface area.


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## guzman (Aug 29, 2010)

StefanPochmann said:


> Using a rectangle, I can get as close to the cube surface area as I want. For example, I can wrap a 10x10x10 using a strip one piece wide and 646 pieces long, that's less than half a side "wasted". In general, for an NxNxN, a strip one piece wide and 6*N*N + 5*N-4 pieces long suffices. For large N, this approaches the cube surface area.



Nice solution !!


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## maggot (Aug 29, 2010)

can you use the T technique? top of T 3*(length of side), bottom of T 3*(length of side), center cube D face on top of T, fold left and right sides to cover L and R face, wrap F, U, B faces with remaining strip from bottom of T. if the paper is slightly over in measurement, say 1-2cm, then the actual area from wrapping with this method would be significantly lower than overlapping from rectangle folding method.


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## Stefan (Aug 29, 2010)

rokicki said:


> you're not allowed to cut the paper ...


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## guzman (Aug 29, 2010)

maggot said:


> can you use the T technique?



I think that 


rokicki said:


> If you're not allowed to cut the paper ...



means that you have to use a rectangular sheet of paper,
to obtain a T you would need to cut your paper.

Stefan's idea of using a long rectangle solves the problem by minimizing (as much as you want) the overlapping areas (overlapping is huge if the rectangle is wide while overlapping can be reduced if you use a narrow rectangle and figure out where to make the folds ... )

Edit: I saw Stefan's last comment after posting mine.


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## qqwref (Aug 29, 2010)

That's a cool solution, Stefan (just what I'd expect of you ) but I can't quite make it work in my head. Maybe the example you chose is just too big. Can you explain your wrapping in more detail?


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## Stefan (Aug 29, 2010)

Zigzag L, then U+B+D+F one slice at a time, then R (see image below). Probably there's a more efficient way, for example zigzagging L+U+R followed by zigzagging F+D+B, but my pattern is enough for the idea to work. Many other paths work as well, for example zigzagging (or spiraling!) one side at a time would lead to about 12*N unit squares overhead which still vanishes compared to the quadratic 6*N*N. When doing one side at a time, you can also go from one side to the next without caring about the overhead, as it's just linear. Or you could go in diagonal layers from UFL to DBR, the possibilities are endless.


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## Stefan (Aug 29, 2010)

Or you could build a fitting T from a rectangular stripe, for example by spiraling like this and wasting only 4*N-1 unit squares.


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## guzman (Aug 29, 2010)

StefanPochmann said:


> Or you could build a fitting T from a rectangular stripe, for example by spiraling like this and wasting only 4*N-1 unit squares.



Nice solution. 
You can also simplify it a tiny bit and waste 4N-2 unit squares.


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