Gifts

[rokicki]

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?

[qqwref]

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.

[guzman]

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.

[Cride5]

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.

[qqwref]

Mm, good point.

[Lorken]

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 anyways)

[rokicki]

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.

[rokicki]

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.

[Stefan]

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.

[guzman]

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 !!