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Given a square piece of paper. Cut it into 4 pieces that could be used to create a right angle pyramid - the 4 pieces are the faces of the pyramid.

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  • $\begingroup$ What's the definition of a right angle pyramid? $\endgroup$ – hexomino Jul 12 at 8:52
  • $\begingroup$ I assume the question means a right pyramid, rather than right angled. A right pyramid has the apex directly above the centre of the base. $\endgroup$ – Showsni Jul 12 at 12:04
  • $\begingroup$ No. Use this definition: " A right-angled pyramid has its apex above an edge or vertex of the base. In a tetrahedron these qualifiers change based on which face is considered the base" and the requirement is - apes above a vertex. $\endgroup$ – Moti Jul 12 at 15:00
  • $\begingroup$ @Moti The pyramid in our solutions is a right-angled pyramid then. If you consider any of the right-angled triangles as the base, the apex will lie above one of the vertices of the base. $\endgroup$ – hexomino Jul 12 at 16:21
  • $\begingroup$ I am seeking a solution where three faces are perpendicular to each other - three edges are along the cartesian axes and 4 faces (I think your solution contains a face that is not a piece of the cut) $\endgroup$ – Moti Jul 12 at 16:33
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I think you could make cuts as follows

enter image description here
so that $a$ and $b$ are the midpoints of the sides of the square.

Then, the three triangles surrounding the central triangle can be folded up along their adjoining edges to create a pyramid.

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  • $\begingroup$ I edited the puzzle to reflect the requirement for a right angle pyramid. Hope I am not disappointing you since you have a nice solution. $\endgroup$ – Moti Jul 12 at 6:54
  • $\begingroup$ I think your solution uses an additional face that is not part of the cuts. $\endgroup$ – Moti Jul 12 at 16:34
  • $\begingroup$ @Moti There are four triangles in the picture. Those are the four faces, why would I need another one? $\endgroup$ – hexomino Jul 12 at 16:37
  • $\begingroup$ @Moti, if we use the small triangle as the base, we could label the coordinates of this pyramid in three-space as (0,0,0), (1,0,0), (0,1,0) and (0,0,2). This is exactly what you've described in your comment. $\endgroup$ – hexomino Jul 12 at 16:40
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    $\begingroup$ You are right and I am accepting your solution $\endgroup$ – Moti Jul 12 at 17:15
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How about this cut? (With Region A as the base of the pyramid)

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  • $\begingroup$ Seems to be exactly the same as the answer that preceded yours by a few minutes. $\endgroup$ – msh210 Jul 11 at 14:56
  • $\begingroup$ It is, I was putting together the diagram and as I was posting I was notified the other solution was posted. I'm fairly certain this is a unique solution anyway. $\endgroup$ – Michael Moschella Jul 11 at 17:48
  • $\begingroup$ There is another solution for a right angle pyramid - see edited puzzle. $\endgroup$ – Moti Jul 12 at 6:52
  • $\begingroup$ See also other comments - your solution is of 5 faces pyramid. $\endgroup$ – Moti Jul 12 at 16:35

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