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The planks all have to be identical. But I'm trying to figure out if there is a configuration in which it can be created. Below is an example of a screwless square shape. This cube needs to be able to be assembled, not just fit together.

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Alanay
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I believe that the answer is no, and though I am not equipped to provide you with a proof, I can give you a rather lengthy explanation.

When you assemble your two-dimensional example in a plane, the entire reason that it is rigid is because it can't be unassembled in the plane, and to assemble it, you must lift the pieces out of the plane, and then place them back down. However, in order to assemble a cube in an analogous way, you need to be able to pull the pieces into the fourth dimension.

As for @RoddyMacPhee's answer, I don't believe that you can assemble the walls of a cube such that each wall is connected to each other wall using dovetails. Such a cube would be impossible to take apart without destroying it, suggesting that it would not be assemble-able.

TLDR: No, unless you are a fourth-dimensional being.

Franklin Pezzuti Dyer
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  • Thank you for the answer, do you think the same applies if we use 2 planks (sides) as one, so there are only 6 pieces that need to be identical and fit together? Or even 3, 4 or 6 sides already connected? I could imagine it being possible if 6 of the sides are already connected by sliding them together diagonally. – Alanay Jul 27 '17 at 21:35
  • Yes, that might work... – Franklin Pezzuti Dyer Jul 27 '17 at 21:38
  • After working on it a bit more in SketchUp I've found it's almost possible, if the cube was made out of rubber instead of wood with flexible properties it could work. http://imgur.com/a/sMPT1 @Nominal Animal: Thanks for your last comment, I hope I could try something like that on a small scale soon. – Alanay Jul 27 '17 at 22:05
  • @Alanay: If the planks in the question refers to a material like pine: It is possible to severely deform the wood by boiling it in water, and then applying compression. After it has cooled down, boiling it again in water will re-form the wood to roughly its original shape. This is used in the construction of many wooden puzzles. Very little deformation is needed to construct a locking structure; usually, boiling in water is not necessary, as wood does flex a little, especially in the transverse direction. – Nominal Animal Jul 27 '17 at 22:05
  • @NominalAnimal Good point, but I was assuming that the cube would be made of a rigid material. – Franklin Pezzuti Dyer Jul 27 '17 at 22:21