What set of rules are there to identify nets of six square that can roll up into a cube?

Asked by Anythingnew (46) October 6th, 2012

The first question is with any net of six squares what rules can I go by to quickly know if it can be rolled up into a cube?

The next question is slightly more difficult in knowing.
Imagine you are on a 6×6 sqaure floor grid
http://imageshack.us/a/img713/5530/rolledupcube.png
Now this is a net already colored out, starting from the blue and folding 90 degrees up towards the next square and following these steps how do I know if this net can be folded into a square?

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Can you muscle it out by saying:
6 in a row does not work
5 in a row does not work
4 in a row works if there is a single square on either side
3 in a row works if there is two in a row on one side and 1 on the other.
2 in a row might be a subset of the others. check it out.
1 in a row does not work.

LuckyGuy (33334)

Welcome to Fluther.

What a great question!

My question in return is: Does the shape have to “roll up” serially (as the pieces in your example), or would you permit a shape so that the base and its four adjacent sides were set out so that the sides could fold up, and one of the sides could also fold over the top? I hope my description was clear. I don’t have the time (or inclination) to attempt to diagram and post that.

CWOTUS (24126)

Nothing with more than three folds in a row can; three folds nets four sides, and anything more leads to overlap.

I will expand later when I have more time to put words to the pictures in my head, but I can say that it’s far easier to figure out I’d you’re autistic; this sort of problem lends itself more towards visual thinking than to written rules. I can see whether or not something will cube by simply folding it in my head, but I’ll be damned if I can explain it.

@LuckyGuy I don’t see three in a row working at all, unless I misread your meaning.

jerv (31025)

I’ve found a solution to the first question
http://www.mymathcounts.com/documents/Chapter15Netsample.pdf

I just need help with the second part

No worries I’ve worked it all out

@jerv
3 in a row works. I should have been more clear. Put 3 in a horizontal row.. From the center one extend 2 boxes up. From either of the end extend one box down. Here is a poor diagram, where Xs are the surfaces you want to keep.
0 X 0
0 X 0
X X X (3 in horizontal row)
0 0 X

LuckyGuy (33334)

@LuckyGuy I thought I misread; what I saw from your words was:

0 X 0
X X X
X 0 X

further proof that a picture is worth (at least) a thousand words.

jerv (31025)

@Anythingnew , Good job of finding the Web site. I did a search and the best I came up with was this book There is apparently a good deal of math behind paper folding.

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