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At $14.50 per hour, he's going to take the shortest route.

This is actually quite fun and simple! Even if the problem and my following explanation look complicated :P

Let's look at the three dimensional case. One can parametrize a 3 dimensional cube as the Cartesian product of intervals [0, 1] x [0, 1] x [0, 1]. This means a cube is a set of points (a, b, c) where a, b and c are real numbers between 0 and 1. The 2 dimensional sides of the cube are then given by fixing one coordinate. That is, the 6 sides are

{0}    x [0, 1] x [0, 1], 
{1}    x [0, 1] x [0, 1], 
[0, 1] x {0}    x [0, 1], 
[0, 1] x {1}    x [0, 1], 
[0, 1] x [0, 1] x {0} and 
[0, 1] x [0, 1] x {1}. 

Now we just start in the middle of a side at (0, 0.5, 0.5). To get to the next side we walk towards an edge (0, 0, 0.5) and then to the middle of the next side (0.5, 0, 0.5). We iterate this process until we run out of sides with a fixed 0, then walk towards a side with a fixed 1 and continue there. That is:

   (0  , 0.5, 0.5)
-> (0  , 0  , 0.5) 
-> (0.5, 0  , 0.5) 
-> (0.5, 0  , 0  ) 
-> (0.5, 0.5, 0  ) 
-> (1  , 0.5, 0  ) 
-> (1  , 0.5, 0.5) 
-> (1  , 1  , 0.5) 
-> (0.5, 1  , 0.5) 
-> (0.5, 1  , 1  ) 
-> (0.5, 0.5, 1  ) 

This path basically spirals around the cube, going through every side only once. Here's a visualization (sorry, I'm no artist :P)

The same procedure works on a 4 dimensional cube or any other higher dimension. For the 4 dimensional cube it goes like this:

   (0  , 0.5, 0.5, 0.5)
-> (0  , 0  , 0.5, 0.5) 
-> (0.5, 0  , 0.5, 0.5) 
-> (0.5, 0  , 0  , 0.5) 
-> ...
-> (0.5, 0.5, 0.5, 0  )
-> (1  , 0.5, 0.5, 0  )
-> (1  , 0.5, 0.5, 0.5)
-> (1  , 1  , 0.5, 0.5)
-> ...
-> (0.5, 0.5, 0.5, 1  )

This works for arbitrary dimension except for the 1 dimensional cube (which is just a line) because the "sides" there are the two end points of the line and not connected at all. Additionally note, that it is never specified how edges count in this problem, whether they somehow count towards a face or whether you're allowed to go back and fourth on edges. You could technically only walk along edges and step into the sides every now and then.

Too many people are obsessing about 4d topology in this thread. The real difficulty in the question is the non -deterministic pathfinding of the ant, in the absence of pheromones.

making sure you cannot solve it, so you are perfect for the job

Sure. Draw the cube for me and I will plot it's path.

Entry level positions to Gregg's (fast food sausage roll chain) require 1000 word personal statements as part of online applications

Choose a starting face and remember it. Walk each face of a cell containing that face touching each face once much like you would a 3-cube.

Pick any adjoining cell and move into one of its faces from there, walk each of its faces saving the one opposite the face you started on for last.

From there you're on a shared face with the cell opposite your starting cell. Traverse this one in a similar manner to the last, but this time also visit the adjoining faces of each cell adjacent to the second cell you filled, before once again ending opposite the face you started on for this cell.

Now you're on a shared face with the final cell, opposite the face you started on. Walk around the remaining four faces and you're done.

Okay, if you can explain to me in detail how four dimensional topology is going to be important to me while I'm stocking the shelves of your grocery store, I'll give you an answer.

This is a lot like when Boston PD was found to screen out all the smart applicants. Sometime the company wants an obedient idiot.

https://www.cbsnews.com/news/too-smart-to-be-a-cop/

Every cube is four dimensional, assuming time as the fourth dimension. So it would travel forward in time at a relatively constant rate (since ants don’t typically walk at relativistic speeds [citation needed]) and it would traverse the other three dimensions in normal ant ways.

Four dimensional? That is a tesseract. This is impossible to describe how an ant would even interact with let alone touch all eight cells only once.

Isn't a cube by definition a 3 dimensional object? If it were 4 dimensional, it would no longer be a cube.