“2048 Tiles” , has recently gone viral on social network platforms. The aim of the game is to make the number 2048 appear on one of the tiles. Personally, I was able to reach 512 quite easily, but haven’t made any progress after that. However, since I have a natural inclination for lower bounds, here is a new (and less time consuming) challenge:
New Challenge: Is it possible to end the game with 4 as the largest number on any of the tiles?*
A lower bound of 4 is trivially true, otherwise we will have only 2’s on the board and this can be merged in any direction. It seems like a possible tight lower bound too. For instance, consider the following board:
2 4 2 4
4 2 4 2
2 4 2 4
4 2 4 2
The game would end in such a scenario, but is such a situation possible?
I think yes. If we just restrict to left-right movement, we can obtain any row of the above board and thus if the random opening of tiles is in our favour, then we get a lower bound of 4. However the probability that this happens is at most .
I played arbitrary moves for 10-20 games, and always ended up with a score of 64/128 (more of 128 and less of 64). If you were generous enough to call my arbitrary actions as random actions, then we know what the average score is and how far it is from the claimed lower bound and upper bound. (Also, can we prove the upper bound of 2048? — I think this should be possible, by accounting for the space needed to create one 2048 number).
EDIT: As discussed in the comments, the upper bound of 131072 seems to be tight. This might provide some motivation to people who have already reached 4096.