The Polydron Enigma consists of a 6 square and 8 triangular Polydron tiles. Polydron is a brand of construction tiles that can connect along their edges to form a hinge. The 14 tiles of the Enigma join together to form a cuboctahedron. Every tile has a path drawn on one side that goes from one edge or corner to another. The aim of the puzzle is to build the cuboctahedron such that there is one continuous path that visits all its faces and which does not visit any corner more than once.
The tiles are shown below.
David Singmaster wrote about tile construction sets in Cubic Circular #7/8 (1985), p20-23, and noted that the Polydron tiles were "definitely the nicest that I have seen." The Polydron Enigma was released a few years later. The tiles were invented by Edward Harvey, and patented as GB 1,378,942, published on 2 January 1975. He also designed the Polydron Enigma.
The puzzle's 6 squares can be ordered in 6! ways, and rotated in 46 ways. Similarly, the
triangles have 8! orderings and 38 orientations. The tiles are all distinct and have not
rotational symmetry. Therefore they can be put together in 6!·8!·46·38 ways.
The cuboctahedron has 24 orientations, so number of distinct completed cuboctahedra that can be
made is 6!·8!·46·38 / 24 = 32,506,616,217,600.
This is of course not a realistic measure of difficulty, since you would not continue placing tiles to the end after one tile is mismatched.
I have run a computer search to find all solutions. It found 218 where all the edges matched, and no corner is visited more than once. Many of these however consist of two or more loops rather than a single long loop. After weeding those out, only 81 single loop solutions remain.