# What's the minimal time needed to get the highest possible sum of tiles in 2048?

I was reading this question / answers realized that people spend a lot more time playing this game than I thought.

I find it interesting - although you may not - to know what is the minimal time needed to get the maximal sum.

Under reasonable assumptions, the question reduces to: What is the minimal number of moves needed to end up with the highest sum of tiles?

(It is still interesting to think about how much time I need to be playing to get the super score! I'm not sure if the game keeps records of times somewhere.)

There is no single anawer to this question because the game randomly adds a 2 or 4 tile. But it is possible to give a range.

This answer shows the maximum possible score. The sum of all tiles is:

$2^{18}-4=262140$

To reach this result you need a 4 as a new tile at least at 16 specific times:

• first time, when the rest of the board is filled with values from 4 to 65536
• second time, when the rest of the board is filled with values from 4 to 32768 and 131072
• and so in until the last 4

Let's assume the rest of the new tiles were always a 2. All values on the board are from the 2 starting tiles (also assumed as 2) and all added tiles. That means, in this case we have the following number of moves:

$\frac{262140-(2*2)-(16*4)}{2}+16=131052$

If we assume we always get a 4 as new tile and also 2 times 4 as the starting tiles it's even simpler:

$\frac{262140-(2*4)}{4}=65533$

Assuming 1 second per move it‘s between 18.2 and 36.4 hours.

• There is a single answer, because it's a minimal time. Assume the game adds the optimal tile in the optimal location every time, and that the player makes the optimal move. – Samthere Nov 2 '15 at 10:53

# 3,932,156 points over 18.2 hours @ 1 move / second

Let's assume 1 move / second and all thanks to Reddit

This assumes that every tile you spawn is a 2, and the last tile you spawn is a 4.

There are $16$ squares. You'd think that the highest possible number would be $2 ^{16} (=65536)$, which would fill up the board, making it impossible to progress. However, if you do manage to spawn a 4 tile as your last tile, then you can continue to progress to $2^{17} (=131,072)$.

Scoring works like this:

Merging two lower-tier blocks together will give you the score of the higher-tier block (score of $+8$ gained from merging two 4s). For any specific tile score, you have to add up all the scores from the lower tiers.

• Creating a 2 tile $= +0$ pts
• Creating a 4 tile $= 4 = +4$ pts
• Creating an 8 tile $= 8 + (2\times4) = +16$ pts
• Creating a 16 tile $= 16 + (2\times8) + (4\times4) = +48$ pts
• Creating a 32 tile $= 32 + (2\times16) + (4\times8) + (8\times4) = +128$ pts ...

Which can be simplified:

• Tile $2^1$$= 0 \times 2^1 = 0 pts • Tile 2^2$$ = 1 \times 2^2 = 4$ pts
• Tile $2^3$$= 2 \times 2^3 = 16 pts • Tile 2^4$$ = 3 \times 2^4 = 48$ pts

Making tile $2^n$$= (n-1)2^n points. Making the max tile 2^{17}$$ = 16 \times 2^{17} = 2,097,152$ points. You would need to spawn 65,536 2 tiles to make a single 131,072 tile, which, at 1 legal move per second, would take 18.2 hours.

That's only making the max tile. Now we still need to fill the rest of the board. (Note: combining times are correct, as when you merge two tiles together, you spawn a 2 at the same time)

• Next tile $2^{16}$$= 983,040 points (2^{15} tiles or 9.1 hours) • 2^{15}$$ = 458,752$ points ($2^{14}$ tiles or $4.55$ hours)
• $2^{14}$$= 212,992 points (2^{13} tiles or 2.27 hours) • 2^{13}$$ = 98,304$ points ($2^{12}$ tiles or $1.13$ hours)
• $2^{12}$$= 45,056 points (2^{11} tiles or 34 minutes) • 2^{11}$$ = 20,480$ points ($2^{10}$ tiles or $17$ minutes)
• $2^{10}$$= 9216 points (512 tiles or 8.5 minutes) • 512$$ = 4096$ points ($256$ tiles or $4.3$ minutes)
• $256$$= 1792 points (128 tiles or 2.1 minutes) • 128$$ = 768$ points ($64$ tiles or $64$ seconds)
• $64$$= 320 points (32 tiles or 32 seconds) • 32$$ = 128$ points ($16$ tiles or $16$ seconds)
• $16$$= 48 points (8 tiles or 8 seconds) • 8$$ = 12$ points ($3$ tiles or $3$ seconds)(This space will be created by two 2 tiles, and the last 4 tile, so $-4$ points from the equation)

This fills up 15 of the 16 spaces on the board. We don't include the last tile, as it cannot merge with the 8.

Max points possible: $3,932,156$.
Max tiles spawned: $131,070$ ($+1$ for last tile, $-1$ for the 4)
Around 36.4 hours @ 1 move per second.

If you happen to miss out on the last 4 tile, your max score would be $1,835,008$, with $65,535$ ($+1$ for last tile) 2 tiles spawned, and $18.2$ hours @ 1 move per second.

• This is going for highest score, not minimal moves to get the highest possible tile sum. Also, it has at least one mathematical error, pointed out in the Reddit comments. – user2357112 supports Monica Oct 28 '15 at 21:57