Here's what I tried:
Let's call the faces of the cube as follows:
Front - 1
Back - 4
Right - 3
Left - 5
Up - 6
Down - 2
I tried to upload an image of it, but Nginx said it was too big for some reason. You can see the image here though: https://ibb.co/GtHW21P
The starting will be face 1. By side, I mean any of the faces 3,6,5,2.
So, as Melody said,
If the second last position (3rd) is front or back then it is impossible. So the 3rd position must be a side.
So, the 3rd position, as in my diagram, must be on faces 3,6,5,2.
We have 2 cases:
Melody did the same thing (so credit goes to Melody). Now we have to calculate the number of possibilities for each case:
But Wait!!! I just realized that we can go back to the front face before the 4 minutes are over!!!
So now we have more cases:
And our possibilities for this case are:
So, our total is 48/256 (Melody found out the denominator) = 3/16.
Thanks Melody. It just occured to me that we could go back to the front face. I got the same answer as you previously, but something struck me when I saw the 4 x 4 x 4 x 4 = 256. I realized that then we were going back faces, so we could go back to the front face. Thanks again