# Maximum tile possible in a game if 2048 on an nXn grid

Playing on an $n \times n$ grid, how can we determine the largest possible title you can achieve in a game, assuming the computer places tiles in the perfect spots.

The answer is $2^{n^2+1}$. This is because $n^2$ is the number of squares in an $n\times n$ grid, and the $+1$ comes from requiring that the last square spawned be a $4 = 2^2$.
• It's easy to see that in order to form a square of value $2^{m+1}$, having already a square of value $2^m$, you will need to fill $m$ squares ($m \geq 2$); of course, this assumes perfect conditions. For instance, in order to form a square of value $8$, having already a square of value $4$, you will need two $4$'s, for a total of two squares. In order to to form a square of value $16$, having already a square of value $8$, you will need two $4$'s and one $8$, for a total of three squares. This goes on, and the maximum number of squares you can use is $n^2$. – Fimpellizieri Oct 28 '15 at 22:23
• That only argues that it's impossible to create a bigger tile. Can you prove creating a tile of value $2^{n^2+1}$ is actually possible? – user2357112 supports Monica Oct 28 '15 at 22:29
• Use the strategy described above, with base case a square $4$ spawned in a corner and going row by row in a zig-zag fashion. – Fimpellizieri Oct 28 '15 at 22:45