This results in the following strange shape (below), which is the key to the entire assembly.
It is in fact half a tetrahedron, and if you took its mirror image it would fit right into it
to make a full tetrahedron.
The balls at the top and bottom are two of the four corners of the tetrahedron.
We now place the final two pieces.
To orient them correctly it helps to remember that the balls at the obtuse ends
become the corners of the tetrahedron.
The final piece is now the trickiest to place, because it must be snapped into place.
Grab it as shown below and slide in the 90 degree end first,
so that its position is as in the second photo below.
As indicated by the red arrow, push this piece into place.
You should feel some resistance as it snaps into position.
Congratulations, you have aseembled the puzzle!
Give it a toss in the air and it should stay together.
After the first piece is loose, the puzzle will fall apart.