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Joined: April 29th, 2016, 1:40 pm

### wrap the cube

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.

outrun
Topic Author
Posts: 4573
Joined: April 29th, 2016, 1:40 pm

### Re: wrap the cube

For those who want to cause trouble:
1) This is a 2d redering of a 3d scene, if you fail to see the 3d scene then this puzzle is not for you.
2) The cube is the orange object, it's orange because it's made of Adamantium which can't be cut, not even by Wolverine.
3) The blue paper below the orange is the thing you need to cut and wrap around the orange cube. If you plan to cut your computer screen then go back and read 1) again.

outrun
Topic Author
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Joined: April 29th, 2016, 1:40 pm

### Re: wrap the cube

I just got a pm and I'd like to add that I'm looking for proof that's not using the axiom of choice nor its negation.

Paul
Posts: 10418
Joined: July 20th, 2001, 3:28 pm

### Re: wrap the cube

For those who want to cause trouble:
1) This is a 2d redering of a 3d scene, if you fail to see the 3d scene then this puzzle is not for you.
2) The cube is the orange object, it's orange because it's made of Adamantium which can't be cut, not even by Wolverine.
3) The blue paper below the orange is the thing you need to cut and wrap around the orange cube. If you plan to cut your computer screen then go back and read 1) again.
What is "underneath" the cube or "behind" it?!

outrun
Topic Author
Posts: 4573
Joined: April 29th, 2016, 1:40 pm

### Re: wrap the cube

For those who want to cause trouble:
1) This is a 2d redering of a 3d scene, if you fail to see the 3d scene then this puzzle is not for you.
2) The cube is the orange object, it's orange because it's made of Adamantium which can't be cut, not even by Wolverine.
3) The blue paper below the orange is the thing you need to cut and wrap around the orange cube. If you plan to cut your computer screen then go back and read 1) again.
What is "underneath" the cube or "behind" it?!
The thing underneath and behind the cube is "the paper".
The paper has a symmetry group D4. Here is a plot of the cycle graph:

and the symmetry matrices:

You can use this symmetry to either accept/reject hypothetical weird shapes you come up, -or-, you can use the D4 coordinate transforms to find bijective mappings between the visible and occluded coordinates and solve it that way. This last method gives you a fast predicable solution. The explorative "hypothizing shapes" gives unpredictable results, maybe you get it right the first time, maybe you'll be guessing shapes endlessly...

Paul
Posts: 10418
Joined: July 20th, 2001, 3:28 pm

### Re: wrap the cube

Ok. Yes!

But can "I" do it?

No!

So the answer is yes and no. Am I right?

(You started it!)

outrun
Topic Author
Posts: 4573
Joined: April 29th, 2016, 1:40 pm

### Re: wrap the cube

I don't know the answer and I forgot where I saw it! I should have read the answer.

Ah, yes, .. of course.. I should have said "is it possible to.." or maybe "can you proof that is possible (or not) to.." or else "can you proof that it can't be proved that .."

I would place my bets on graph theory. We have vertices, shared edges, faces.. Should be easy.

outrun
Topic Author
Posts: 4573
Joined: April 29th, 2016, 1:40 pm

### Re: wrap the cube

Wait a minute, The surface area of the cube is 6, the paper 9, that's too much! This makes is lot more complicated.

From a psychological point of view, wouldn't it be great is someone made this puzzle and the answer was "no, can't be done." ?

Paul
Posts: 10418
Joined: July 20th, 2001, 3:28 pm

### Re: wrap the cube

Wait a minute, The surface area of the cube is 6, the paper 9, that's too much! This makes is lot more complicated.
Next question: Maximize the number of leftover squares.
From a psychological point of view, wouldn't it be great is someone made this puzzle and the answer was "no, can't be done." ?
I did this once as part of a marketing plan for the magazine. We printed cards with lots of these puzzles: "Move one match to..."; Magic Eye; "Turn over three coins to...", etc. My brilliant, if I say so myself, idea was by making the puzzles impossible people would stare at the card (and magazine logo and website) for a long time. The tagline was "Do you get it?" and obv has many interpretations. God, I used to be soooo good! When did it all go wrong?

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Topic Author
Posts: 4573
Joined: April 29th, 2016, 1:40 pm

### Re: wrap the cube

Hahaha, that's horrible!! I bet some are still staring at it!

I try these things constantly on my kids of course. About the puzzle: I've figured out it's impossible! Ha!

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Topic Author
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### Re: wrap the cube

I was wrong! It's possible!

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### Re: wrap the cube

LOL! Tricky! But DONE!

Make a spiral of the 9 squares but cutting in from the edge for 2 squares, turning, cutting 1 square, turning, and cutting 1 square to have the center square cut on 3 sides.

Starting at the center, crease the center square 90° toward it's neighbor, that square 90° to it's neighbor, and that square 90° to it's neighbor to form an open box shape that's missing 2 sides.

Next (and the likely "out-of-the-box" thought that most miss) is to crease the next neighbor 180° back on the developing box.

Then crease the next neighbor 90° to cover one of the remaining open sides, then make another 180° crease, then the final 90°.

outrun
Topic Author
Posts: 4573
Joined: April 29th, 2016, 1:40 pm

### Re: wrap the cube

LOL!  Tricky!  But DONE!

Make a spiral of the 9 squares but cutting in from the edge for 2 squares, turning, cutting 1 square, turning, and cutting 1 square to have the center square cut on 3 sides.

Starting at the center, crease the center square 90° toward it's neighbor, that square 90° to it's neighbor, and that square 90° to it's neighbor to form an open box shape that's missing 2 sides.

Next (and the likely "out-of-the-box" thought that most miss) is to crease the next neighbor 180° back on the developing box.

Then crease the next neighbor 90° to cover one of the remaining open sides, then make another 180° crease, then the final 90°.
Congrats!!
This sounds *a lot* easier than the dance I once did trying to impress my future wife!

Paul
Posts: 10418
Joined: July 20th, 2001, 3:28 pm

### Re: wrap the cube

LOL! Tricky! But DONE!

Make a spiral of the 9 squares but cutting in from the edge for 2 squares, turning, cutting 1 square, turning, and cutting 1 square to have the center square cut on 3 sides.

Starting at the center, crease the center square 90° toward it's neighbor, that square 90° to it's neighbor, and that square 90° to it's neighbor to form an open box shape that's missing 2 sides.

Next (and the likely "out-of-the-box" thought that most miss) is to crease the next neighbor 180° back on the developing box.

Then crease the next neighbor 90° to cover one of the remaining open sides, then make another 180° crease, then the final 90°.
I had one cut, two edges long.

How many leftover squares? I had one double layer and two leftovers.

Posts: 23951
Joined: September 20th, 2002, 8:30 pm

### Re: wrap the cube

LOL! Tricky! But DONE!

Make a spiral of the 9 squares but cutting in from the edge for 2 squares, turning, cutting 1 square, turning, and cutting 1 square to have the center square cut on 3 sides.

Starting at the center, crease the center square 90° toward it's neighbor, that square 90° to it's neighbor, and that square 90° to it's neighbor to form an open box shape that's missing 2 sides.

Next (and the likely "out-of-the-box" thought that most miss) is to crease the next neighbor 180° back on the developing box.

Then crease the next neighbor 90° to cover one of the remaining open sides, then make another 180° crease, then the final 90°.
How many leftover squares? I had one double layer and two leftovers.
The solution I described has no leftovers -- there's one overlap early in the folding and then each 180° crease creates an overlap.

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