In a 2048 or Threes-like game, what is the highest possible score?

October 6, 2014 8:00 PM

In one of these puzzle games like 2048 or Threes, where you merge tiles to create higher numbers, is there any way to prove what is the highest possible tile number that can be created given the rules and dimensions?

For example, if the 2048 game was played in a 3x3 space instead of a 4x4 space, would it still be possible to create 2048 in that space? If it was played in a 5x5 space, how much higher could we theoretically go? 8192? 32768?

Is there a way to figure out upper and lower bounds on different playfields?

posted by RobotHero to Grab Bag (8 answers total)

Response by poster: Actually, now I've searched properly for this and there's some discussion here and here that I'm trying to work through.
posted by RobotHero at 8:08 PM on October 6, 2014

Not a proof, but this link might be of interest to you also.
posted by forthright at 8:12 PM on October 6, 2014

Response by poster: Okay, the hypothetical chain of descending pieces winding across the board, that does make sense as an upper bound. So the highest number a tile could be merged in an N-tile playfield would be 2^(N+1)? And the +1 I think comes from the possible random initial tiles. If we only got 2s, then 2^N would be the max, and if we could get 8s, then we could get up to 2^(N+2).
posted by RobotHero at 8:56 PM on October 6, 2014 [1 favorite]

I think (as you say) the answer depends on what tiles are generated (or even can be generated) each turn as you get to higher levels.

It should be relatively easy, though tiresome, to demonstrate what the highest possible score is -- draw yourself a board and choose the most optimistic tile generation and placement each turn and see how high you can get.
posted by Nerd of the North at 9:23 PM on October 6, 2014

Here is a discussion of the basic idea of an inductive proof to show that in 2048, given the most ideal drops of the lowest value tiles, the highest achievable score is 2^k, where k is the number of positions in your grid.
posted by The Great Big Mulp at 9:57 PM on October 6, 2014 [4 favorites]

Come back and play for the 65536 tile!
posted by garethspor at 10:48 PM on October 6, 2014 [1 favorite]

Threes generates tiles slightly differently than 2048, allowing a theoretically lucky and skilled player to score almost arbitrarily high through lucky "Bonus Tile" generation. Henry Mercer has built an AI to play Threes, and along with it has posted the tile generation algorithm.
posted by LiteS at 7:34 AM on October 7, 2014

Oh, and, yes, RobotHero, you are correct.
posted by The Great Big Mulp at 8:49 PM on October 7, 2014