I can do it in
The first thing to notice is that each stone takes 2 seconds to move to the adjacent cell. Even if you're moving 2 stones at once, it still takes 2 seconds per stone to move. So, to figure out the best cell to move them all, all you have to do is:
Find a cell that will take the least amount of moves to get all the stones to. Since the board is symmetric, you only need to consider 4 cells:
012 101 123 212
123 212 012 101
234 323 123 212
345 434 234 323
Total moves: 30 26 24 20
The last configuration only requires
20 moves - 40 seconds total, so that's the best one. Notice that since it's symmetric any one of the 2 middle cells can be used.
40 seconds is the least amount of time to actually move the stones. To minimize the total time it takes, you have to minimize the time it takes you to go get the stones from the final cell. Going to get one stone at a time will not work, because you will need to go out more often. So you have to make sure to bring back at least two stones every time you go out. You have 11 stones to collect, and can only carry 2 stones at a time, so you'll need to leave at least 6 times. Here's what the times look like to get to the stones to bring them back:
As you can see
I x'ed out the second stones that you will pick up on the way back from each outing. So the final time to get the stones is 13 seconds.
But wait! You don't actually start in the final cell, you start in the corner. That means you can eliminate one of the times that it takes you to get to the corner from the final cell. The other answers here decided to eliminate the 2 second corner (for some reason? LOL, I only jest!), but if you eliminate the 3 second corner, the total time will be 40 + 10 = 50 seconds!
Ok, to make things clear, here's the final solution:
You are in a corner. Grab the stone, move up, grab another stone, move up then move right. Time: 10 seconds.
Go down, go down.Time: 2 seconds.
Grab stone, move up, grab stone, move up. Time: 6 seconds.
Go down, go down, go right. Time: 3 seconds.
Grab stone, move up, grab stone, move up, move left. Time: 10 seconds.
Move up, move left. Time: 2 seconds.
Grab a stone, move down, grab a stone, move right. Time: 6 seconds.
Move up, move right. Time: 2 seconds.
Grab a stone, move down, grab a stone, move left. Time: 6 seconds.
Move up. Time: 1 second.
Grab a stone, move down. Time: 2 seconds.
Total time: 10 + 2 + 6 + 3 + 10 + 2 + 6 + 2 + 6 + 1 + 2 = 50 seconds.