Number of moves to beat 2048 activity

I'm early on in developing this activity for my class and who knows--I may even be thinking about it incorrectly but what I do know is this game is both addicting and very relevant to my students right now. What I'm considering posing to them as a question in our exponent unit is:

"What is the best and worst case scenario of the amount of time I would need to 'beat' the game and get to the 2048 tile if I take 5 seconds with each move?"

If you don't know the game you can check out this video but I'll warn you--while you are playing this game life happens around you without you being aware of anything going on (very addicting).

During the game you continue to get new 2 and 4 tiles that you must push together to try and build up to a tile of value 2048 (2¹¹). In a worst case scenario you would continue to get all 2's as new tiles or best case scenario all 4 tiles (neither happens but it sets up a range of what could).

Here are some scenarios that could give you a range of values for acquiring the 8 tile and the 16 tile:

What I like about the idea of this problem is there is real motivation in wanting to get to an equation but the student will need to be sure that the range of moves can all be generated. I haven't presented it to students yet so more may be following on how that goes.

I will leave the 4 tile graphic out to allow students the opportunity to consider it on their own and maybe I won't even present them with these scenarios early on in the problem either. Comments or suggestions welcome as always in the comments below. If there is a better example of this somewhere else too please let me know!