Maximum board position in 2048 game A game called 2048 is making rounds on social media. I am trying to determine the maximum score attainable for this game.  Let's assume WLOG that only 2s are returned (if 4s are possible the max score is doubled).

First, I'll relax some of the rules to create a game, $\mathcal{G}$, that can achieve a score at least as high as the original, 2048 game.  A run $R$  of $\mathcal{G}_n$ consists of a sequence of moves $m_1,\ldots,m_k$ which manipulate a multiset, $\mathcal{S}$ of maximum size n.  Each $m_i$ is either:


a put $p$, which inserts a 2 into $\mathcal{S}$, or
a merge, $\alpha_i$, which replaces two $2^i$ elements from $\mathcal{S}$ with a single $2^{i+1}$ elt.


I allow sequential moves of the same type because 2048 allows multiple merges or no-merge turns.  The score of a position is the sum of $\mathcal{S}$.

What is the maximum score possible on a game of order n?

Note: The final position must have only unique values, else a merge and put would be possible in increase score.

I claim it is $\sum_{i=1}^{n} 2^i$, but couldn't come up with any good induction arguments.  It's pretty easy to prove that an element $2^k$ is producible in a game $\mathcal{G}_k$, but proving an upper bound for production got a bit hairy, and also doesn't cover the full space.
An optimum board just before the last number is played on 4x4:

$$\begin{array} {|c|c|c|c|} 
\hline
&amp; 256 &amp; 512 &amp; 65536 \\ \hline
4  &amp; 128 &amp; 1024 &amp; 32768 \\ \hline
8  &amp; 64 &amp; 2048 &amp; 16384 \\ \hline
16 &amp; 32 &amp; 4096 &amp; 8192 \\ \hline
\end{array}$$

... and then a 4 luckily falls into the empty square giving the largest number that can be created is $2^{17} = 131072$ .  The fact is, in order to make a number $2^n$, then you have to have all the numbers $[4 \dots 2^{n-1}]$ already present on the board.

The largest number that can be made is then $2^{\left(n^2 + 1\right)}$, or if you assume that no 4s will fall, then it is $2^{\left(n^2\right)}$.

Such a board position can be easily constructed by always having an increasing sequence in the nonempty squares of the path starting at the top left:

$$\begin{array} {|c|c|c|c|} 
\hline
\downarrow &amp; \rightarrow &amp; \downarrow &amp; \circ \\ \hline
\downarrow &amp; \uparrow &amp; \downarrow &amp; \uparrow \\ \hline
\downarrow  &amp; \uparrow  &amp; \downarrow  &amp; \uparrow \\ \hline
\rightarrow &amp; \uparrow &amp; \rightarrow &amp; \uparrow \\ \hline
\end{array}$$

and then having the next number fall on the last empty square of the path.

Aside, this turns out to be a very effectively strategy for playing the game with non-optimal drops as well.

Edit: James Grime to the rescue again.  He made a fairly entertaining youtube video to address this question.
For highest score i calculated 3,932,060 , largest tile 131072 &amp; for a game without 4's appearing it would be 1,834,972 largest tile 65536