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Paul
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Re: wrap the cube

April 12th, 2017, 12:23 pm

When cubes are flat packed they always have four in a row then one square on each side. (Admittedly this allows for tabs for connecting sides.) So they fill a three by four rectangle.

Are these different flat packs more efficient in any way? Obviously they take up 3x3. But what about tiling the flat packs?
 
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Re: wrap the cube

April 12th, 2017, 12:38 pm

That tiling an floor with these tiles?
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Re: wrap the cube

April 12th, 2017, 12:51 pm

This one below can be folded into a 4d hypercube, but what about cutting a 2x2x2 stack of 3d cubes? Note that a 2x2x2 can't have overlapping cubes.

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Traden4Alpha
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Re: wrap the cube

April 12th, 2017, 12:56 pm

When cubes are flat packed they always have four in a row then one square on each side. (Admittedly this allows for tabs for connecting sides.) So they fill a three by four rectangle.

Are these different flat packs more efficient in any way? Obviously they take up 3x3. But what about tiling the flat packs?
Topologists might make cubes that way but not logistics packaging makers due to the added performance requirements on real-world boxes in conjunction with the anisotropic nature of corrugated cardboard and anisotropic nature of the world (gravity).

It seems more common to use a 1x4 strip for the four sides flanked by 1/2 x 1 flaps on both sides of that strip. That uses a total of 2x4 units of material (plus a little for a flap to hold the sides together) but it also ensures that the orientation of the corrugation can support stacked boxes, offers a double-thick bottom for damage resistance, has four folded sides on top for better stacking strength, and is perfectly tiled from a larger roll of corrugated stock.
 
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kermittfrog
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Re: wrap the cube

April 12th, 2017, 5:36 pm

Spiral method here - no leftover tiles.
 
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katastrofa
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Re: wrap the cube

April 12th, 2017, 8:26 pm

Can you wrap this cube?
Here are the rules:

1. The paper may be only cut or folded along the crease lines.

2. The cutting should not cause pieces to separate.

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Re: wrap the cube

April 12th, 2017, 8:39 pm

katastrofa,  what happens with the top left 2x2? A diagonal fold somewhere?
 
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katastrofa
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Re: wrap the cube

April 12th, 2017, 8:47 pm

katastrofa,   what happens with the top left 2x2? A diagonal fold somewhere?
No. Folded along the crease between (1,2) and (1,3) and the two squares glued together. Row-wise layout.
 
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Re: wrap the cube

April 12th, 2017, 9:01 pm

katastrofa,   what happens with the top left 2x2? A diagonal fold somewhere?
No. Folded along the crease between (1,2) and (1,3) and the two squares glued together. Row-wise layout.
Hmm.. a 2x2 can either be folded horizontal  or vertical along its middle, ..but not both!
I think I'm going to cut your example IRL to see what happens!
 
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katastrofa
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Re: wrap the cube

April 12th, 2017, 9:45 pm

Maybe I don't get the rules, because it's the first obvious solution that came to my mind. Anyway, you made me doubt it and I've checked it too :-)
My hands look horrible - I should probably start using the dishwasher :-(
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Re: wrap the cube

April 12th, 2017, 9:48 pm

Very nice and clear Kata!
 
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Paul
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Re: wrap the cube

April 12th, 2017, 9:53 pm

I'm with katastrofa. The rules are vague on the definition of "wrap."

If allowed leftovers then I think you can do it with cutting just two edges.
 
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Paul
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Re: wrap the cube

April 12th, 2017, 9:56 pm

I used to have some nice pastel colour nail varnish.
 
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Re: wrap the cube

April 12th, 2017, 10:40 pm

One rule is that the paper needs to stay connected. I've wrapped presents like that -folds sticking out- (in the wife's shop). I think it's nice.

yes the diswasher: Bułka z masłem!
 
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wileysw
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Re: wrap the cube

April 17th, 2017, 10:07 pm

here is a reference for people who are interested.

paper-folding is one of Martin Gardner's favorite topics (e.g., hexaflexagons) and he discussed a restricted version of OP's problem in his "Mathematical Games" column (Scientific American 202, 161-170, 1960).  below is a screenshot from one of the books that collect his columns:

Image
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