I refer to the wooden snake puzzle in this post:
I notice that there are 6 solutions to the 4x4x4 snake, which start respectively as follows:
Is there any logic or rationale behind the fact that the third step is always the opposite direction of the first step? Why do the other starting steps (e.g. FRU, RFU, UFR) fail to solve the puzzle eventually?
The puzzle essentially has a unique solution. The 6 solutions you list are simply the 3 rotations around the starting corner, together with their mirror images.
So really what you are asking is, why is the solution unique? That is not an easy question to answer in a way that is meaningfully different to "just because it is". Maybe you could look at the three long straight sections of the snake (which are bars that are four cubes long), and see all the places they could be located relative to each other. This cuts down the possibilities. Finding available places for the shorter straight sections cuts it down even further so that there essentially is only one way for the snake to fit into the cube.
I understand that this is unsatisfactory. I have little doubt that the designer of this puzzle purposely chose a combination of bends and straights that happens to have only one solution.
