+118609 Don't know... just thinking. Lets say we start on the front.
4 minutes = 4 moves.
If the second last position (3rd) is front or back then it is impossible. So the 3rd position must be a side.
So the only posibilities that work are
front side back side front 4*1*4*1 = 16ways
front side side side front 4*2*2*1 = 16 ways
So there are 32 paths that end on the start face.
How many paths are there altogther? Each side shares and edge with 4 other sides.
so that is 4*4*4*4 = 256
So I think the probability that the ant will finish on the face where it started is 32/256 = 1/8
+118609 Have you tried to examine my answer to find the floor in my logic or is it more your style just to get everyone else to do you thinking for you ?
I would not answer most people in this way. I am more than happy to own my errors.
But you are just using this place as a cheat site.
+936 I'm sorry. I'm not trying to cheat or anything. This isn't even related to school. These are just questions from competitions and I'm trying to improve my score, so that I can do well when I actually have to do the competition. I appreciate your help, but please don't think I am cheating or trying to avoid trying questions. As you can see, I am answering other people's questions too. When I was telling the correct answer, I wasn't doing it to be mean or something, I was just saying that there is a different answer that is claimed to be the "correct answer". Also, I wouldn't have gotten this far if I made people think for me.
+936 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 
Can anyone solve this?
