2048 is a single-player online and mobile game in which the objective is to slide tiles on a grid to combine them and create a tile with the number 2048. It is played on a simple gray 4×4 grid with tiles of varying colors overlaid that slide smoothly when a player moves them. If two tiles of the same number collide while moving, they will merge into a tile with the total value of the two tiles that collided. Higher-scoring tiles emit a soft glow. Every turn, a new tile (with a value of 2 or 4) will randomly appear in an empty spot on the board. When the player has no legal moves (there are no empty spaces and no adjacent tiles with the same value), the game ends.

If the largest possible tile value that can be achieved in the game is $a^b$ , where $a$ is a prime number and $b$ is a positive integer, find the value of $a+b$ .

Details and assumptions:

Assume the player is skillful enough and keep playing the game after they reach the target tile value, 2048.

The tiles that appear automatically on the board are all with a value of 2 or 4 only.

17
16
18
19

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To get $2^{n}$ , you need two cells of $2^{n-1}$ . Using this recursion, start with two 4's. This adds to 8, so there should be another 8. Continuing in this fashion, you can get $16, 32, 64, ..., 2^{13}, 2^{14}, 2^{15}, and 2^{17}$ , but not higher. In an attempt to get a higher power( $2^{18}$ ), the board will be filled with one of each tile( $2^{2}, 2^{3}, ..., 2^{17}$ ). Thus, the highest it can go is $2^{17}$ , so $a = 2$ and $b = 17$ so $a + b = \boxed {19}$ .