I have four equilateral triangles. I place one on the floor. I pick a random edge of this first triangle and attach it to a side of a second triangle. Next, I randomly pick one of the four edges of the resulting rhombus and attach the third triangle. Finally, I randomly pick an edge from along the perimeter of the resulting shape and attach the fourth triangle.
What is the probability that I can create a regular tetrahedron by folding the four triangles along their edges?
Extra credit: Instead of using four equilateral triangles to make a tetrahedron, suppose I use six squares to make a cube. What is the probability I can make a cube by randomly attaching the squares, one at a time? (And what are my chances of making any of the three other Platonic solids using their respective faces?)
Three randomly connected equilateral triangles, attached by their sides will always yield a trapezoid in two dimensions. When folded, two sides will attach to each other, leaving three for the remaining face.