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Comment by cousin_it | Hacker News Reader

Another nice puzzle:

A cube looks like a square from three orthogonal directions. A cylinder can look like a square from infinitely many directions, but they are all coplanar. Can you find a convex shape that looks like a square from more than three directions, without all of them being coplanar? In particular, can you find a convex shape that looks like a square from two distinct sets of three orthogonal directions? Can you find all such shapes?

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(Rot13 as not to spoil the riddle)

Ubj nobhg n grgenurqeba? Begubtencuvp cebwrpgvba nybat gur yvar pbaarpgvat nal rqtr zvqcbvag gb gur bccbfvgr rqtr zvqcbvag tvirf n fdhner, lvryqvat 3 fhpu cebwrpgvbaf (cyhf gurve bccbfvgr pbhagrecnegf). Abg fher vs gurer ner bgure fhpu funcrf (vtabevat gevivny zbqvsvpngvbaf bs gur grgenurqeba) - qvq lbh znantr gb svther guvf cneg bhg?

A tetrahedron looks like a square from three orthogonal directions, same as a cube. It's possible to get more.

I'm thoroughly impressed by your geometric abilities! I didn't know that, and took me a while to check. Any hints on the puzzle, and as a sidequestion, what tools do you use to figure this kind of question? Just imagination, vector algebra, elementary trigonometry?

Huh? There's nothing to be impressed about. You can stick a tetrahedron inside a cube so it creates the same square shadows in all three directions: https://i.stack.imgur.com/oAUnH.gif

I know a lot of math, but for this puzzle, drawing stuff on paper is enough. Here's a hint: if you cut off one corner of the cube, all shadows are still square. How much can you cut? Can you cut some corners strategically to make at least one new square shadow while keeping all the old ones? How many square shadows can you get?

Do mathematicians consider this an open problem or does there exist a well known solution? And what about a name for this problem, does it have one already?

Nah, it's nothing as grand as that. I saw it on facebook a few months ago and solved it in about ten minutes, just wanted to share the fun.

...do you know the story of George Dantzig?

https://en.wikipedia.org/wiki/George_Dantzig#Mathematical_st...

I don't mean to say that it seems applicable here - but I also don't mean to say that it doesn't seem applicable here.

Orthographic or perspective projection?

I only know the solution for orthographic. But it seems like perspective would be too easy, just make any shape with many square sides, and put the cameras close to the sides.

Cylinders only look like squares in orthographic projection, so my guess is that that's the intention.

Am I allowed to invoke hypercubes because that would cut this knot nicely.

An octahedron?

An octahedron looks like a square from three orthogonal directions, same as a cube. It's possible to get more.

Is the question posed only in relation to three dimensional space?

Yeah, it's the kind of shape you can carve from a block of cheese in a minute.

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