“Tetrakaidecahedron” is a slightly more impressive word for “tetradecahedron” — a polyhedron with 14 faces. You’ve probably never heard the term before. It isn’t much used because it is ambiguous: there are a whole lot of 14-faced polyhedra: 1,496,225,352 to be precise. Among them are the cuboctahedron, the truncated cube and the truncated octahedron. The Tetrakaidecahedron puzzle is an example of the last of these.
The truncated octahedron is closely related to the cube and octahedron, and has the same symmetries. We would therefore expect the Tetrakaidecahedron to have the same core as one of the cube puzzles, and hence present as a similar puzzle.
That is indeed the case. The turning axes of the Tetrakaidecahedron correspond to the four axes of 120° symmetry of the cube / octahedron / truncated octahedron. In a truncated octahedron these axes pass through the centre of opposite decagons, but in the cube they pass through opposite vertices. The cube puzzle that makes use of the same axes of symmetry, with an identical cut, is the Skewb.
The Tetrakaidecahedron is therefore almost identical to the Skewb. One solves it by applying the same algorithms as one uses to solve the Skewb, remembering that square faces on the Tetrakaidecahedron correspond to square centres on the Skewb; and triangle faces on the Tetrakaidecahedron correspond to vertex cubies on the Skewb.
The only difference is the requirements for orienting cubies. On the Skewb, the square centres take a single colour, so orienting them is of no importance. On the Tetrakaidecahedron, the square face cubies contain slices of their neighbouring faces, so orienting them correctly is critical. The opposite applies for the triangle faces. On the Skewb these are three-coloured vertex faces, and must be oriented correctly. On the Tetrakaidecahedron they are uni-coloured triangles, so orientation matters not.
There is another Skewb variant that has exactly the same orientation requirements: the octahedron-shaped Skewb Diamond. As noted, the Tetrakaidecahedron is a truncated octahedron, so we can consider this puzzle to be a Skewb Diamond with the vertices truncated.