4
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With any triangle, 8 copies can make an octahedron. With any acute triangle, 4 copies can make a tetrahedron.

Make a scalene octahedron, then construct scalene tetrahedra on each face. Here are samples with the 4-5-6, 6-12-13, 9-11-13, 11-12-15, and 11-13-16 triangles.
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These are cases where we almost get a 16-triangle outer shell. None of them quite works, the outermost vertices don't actually coincide in these cases.

Is there an integer triangle that gives a perfect solution?

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  • $\begingroup$ At first congruent triangles make tetrahedra of 4 equal face. Although you try to make concave hexahedron on each face, such a shape doesn't exist. en.wikipedia.org/wiki/Hexahedron. $\endgroup$ – Takahiro Waki May 7 '17 at 0:59

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