# Which 3D shape can you make out of this?

The above shape can be folded into a closed 3D shape using no more than 14 distinct folds, with no parts overlapping. What is special about the shape that results?

Rules and clarifications:

• Every part of the black boundary should meet up with another part of the black boundary

• No parts of the paper overlap or touch one another except where they meet at the black boundary

• The shape is made up of flat surfaces

• No cutting except for cutting it out along the solid black lines

• The dotted lines are just to show you how the shape is constructed - they don't necessarily indicate where to place folds.

• You can decide for yourself whether printing it out and trying to fold it is cheating.

I came up with this the other day. I'm not 100% sure the solution is unique, but I think it should be.

• Can the surface have smooth curves, for example as found in a in a cylindrical surface, or are the faces all required to be flat (except where the folds are obviously), for example as in a polyhedron? – Penguino Feb 27 '15 at 5:40
• @Penguino all the surfaces should be flat. (If there's a solution with curved surfaces I'd like to know it though!) – Nathaniel Feb 27 '15 at 5:54
• can you make cuts in the shape? (without dividing it in multiple pieces) – Ivo Beckers Feb 27 '15 at 13:20
• @IvoBeckers no, folds only (apart from cutting out along the black outline, of course.) – Nathaniel Feb 27 '15 at 14:08
• I can make a hat, I can make a broach, I can make a pterodactyl! – KSmarts Feb 27 '15 at 18:54

If folded as shown below, it can be closed along the matching letters, to form a 12 sided polyhedron with six quadrilateral faces and two sets of three isosceles triangular faces. The quads are arranged in two triplets that join to form two right corners (as found in a cube), and the triangles 'join' the two groups of quads together. The shape has 120 degree rotational symmetry around an axis that passes through the two 'cube-corners'.

The shape is a bit difficult to describe, so I have sketched approximate views of it from two different angles to show, very approximately, how it sticks together. All the faces are flat as required.

OOPS!!! - My solution is wrong. To fold into the shape I have described, the long edge of the small triangle (and short edge of the long triangle) would have to be of length 2/sqrt(3), not sqrt(2) as it is in the 'cutout'.

• This is 11 folds. Shouldn't it be 14? – Enigma Mar 1 '15 at 10:28
• It is requested "no more than 14". – Florian F Mar 1 '15 at 10:42
• First, nice 3D representation. But I am afraid it doesn't work. If you add in the small triangle betwee lines a and b, you have a flat surface with 4 folds joining. These can't all be mountain folds. – Florian F Mar 1 '15 at 10:47
• Ah, very good, yes - my solution was not unique after all. I think I will have to give you the check mark and post my own solution as a separate answer. – Nathaniel Mar 1 '15 at 10:54

I can do it with 11 folds also.

• Very nice - we have another winner. – Nathaniel Mar 1 '15 at 11:39
• ...though actually, aren't you using 11 folds? In the top picture it looks like 9, but in the bottom picture you seem to have made two additional folds, one down the centre of each of the two large triangles, and I can't see how it would fit together unless you did that... – Nathaniel Mar 1 '15 at 11:57
• Yes, I miscounted. 11 it is. – Florian F Mar 1 '15 at 15:42

This is the solution I intended, though I've accepted Penguino's Florian F's since it has the smallest number of folds.

Dashed lines are mountain folds, and dotted lines are valley folds.

Perhaps I'll make a 3D representation later, but the special thing about the resulting shape is that it can be made by cutting a cube into two halves as below, rotating one half by 30°, and sticking them back together. [image source]

If anyone can think of a good way to change the puzzle so that this is the only solution, please comment!

• Have you actually folded this? I don't understand how you can use the small squares in the flat paper to represent the small triangles in the 3D...? – Enigma Mar 1 '15 at 11:17
• @Enigma yes, I've folded it. The small squares are each divided in two by valley folds to make the triangles. (Note that the 3D image in my post is not the shape you get from folding it, it's just to show how the shape is constructed. In the actual shape the small triangles are next to each other.) – Nathaniel Mar 1 '15 at 11:40