# 2048

What’s the minimum score you could get to win the game? What’s the maximum? Answers here.

UPDATED BELOW

Most people I know have played this game. It’s a simple game. It is also addictive. (It’s been clinically proven to be second only to crystal meth… Now that isn’t true of course; my friends from the humanities gave it up quite easily*. I kid, I kid.) Go ahead, give it a go. There are also 3D, 4D, and tetris versions of the game if you care to look for them. The 3D version will be decidedly easier to win than the 2D version, of course. Just harder (and more fun) to visualise. I daren’t start with any of these; I’ve wasted enough time as it is.

The object of the game is to get to the 2048 tile and the “You Win” banner. But I thought I’d put down some perhaps interesting things I worked out about the game.

For instance, you must, if you’ve actually finished the game, know that you can play on past 2048. What do you think is the highest tile you could get to on the 4×4 board?

If you said 65536, like so:

You would be almost correct, unfortunately. You see the game is so designed that both 2s and 4s can be spawned. Which of course means that this can happen:

giving a maximum tile of 2^17, or 131072.

Now, getting your tiles to arrange themselves this way would take more than insane concentration and forethought. It would take the computer conspiring with you to give you tiles in just the right order and at just the right place and so on. The maximum of 2^17 is only a theoretical maximum, then. What the maximum is, when the computer isn’t working to help you, is a much harder thing to calculate. I’ve myself never managed to get a tile higher than 2048, even when things looked promising.

You may have noticed that the game also scores you. The scoring is again relatively simple. For every “merger”, you get as many points as the face value of the tile you just created by merger. Spawned tiles don’t add points. Your score is a measure of how inefficiently you’ve played the game; the ideally played game would leave just one tile of value 2048 and one newly spawned tile on the board.

What then, do you think, is the best (read lowest) score you can theoretically get on your way to a winning game? And what’s the maximum score you can theoretically get as you win the game? This isn’t exactly hard to do, but requires some explanation. The score accumulated as you get to a tile of a certain power of 2 is proportional to both the value of the tile and the exponent. The answer also, of course, depends on whether 2s or 4s are spawned.

Assuming it’s all 2s, for instance, the score will be 8*(3-1) = 16 for a tile of 8. For a tile of 16, the score should be 16*(4-1) = 48. If it’s all 4s that are spawned, the score will be 8 and 32 respectively.

The answers to the maximum question I get are 92164 and 83976 depending on whether exclusively 2s or 4s are spawned. The minimum is either 20480 or 18432 respectively.

My average score for winning games hovers around 23000-25000.  CORRECTION: 23000-25000 is my average score at the end of the game. My average score as I win the game is about 20500. What’s yours?

The one question I would really like to answer but can’t is what happens when a monkey plays the game. This question has proven itself beyond me. Maybe somebody reading can help me out.

UPDATE

*It was pointed out to me that this dig at the humanities is uncalled for. The joke isn’t funny or clever. As stereotypes go, the science-vs-humanities divide is a silly and pernicious one. The innumeracy of people from the humanities is no more a fact than that scientists can’t be good writers or speakers, or any other such similar nonsense. I’ve left in the offending sentence; this may, however, be considered a retraction.

## 14 thoughts on “2048”

1. Thanks, Raziman. What’s the average, do they say? I can’t open quora.

But the analytical answer has to be more interesting, no?

2. The expected value of the largest cell seems to be 107.32 (from 5 million simulations) when the next move is made randomly. If the random key pressed is chosen to be different from the previous keypress, the value seems to increase to 121.23.

3. Very nice. Thanks very much. The curve looks like a Poisson, wouldn’t you say?

4. Tried a fit. Doesn’t look Poissonous at all.

5. I got to 2048 with a minimum score of 20192. The values of the rest of the other tiles added up to 44.

6. That’s brilliant, dude. Very efficient.

7. Kbig says:

I won with 20124 pts (first time ever) and went on to get a high score of 32376

8. nosnhoj13 says:

I’ve beat the game many times and my highscore is 33872

9. akubas says:

my highest score at the moment is 38100…getting 2048 is easy now, but it kinda pisses me off that I could not get beyond that. The highest configuration I’ve managed is one 2048, one 1024 and the rest of others. At this point I kinda doubt 4096 actually can be done. Does anyone of you actually witnessed or managed to do so. I just wish no one ever did so I can just forget about it…please do tell…

10. I get to 2048 regularly…but the game ends and I can’t continue to score. Videos I’ve seen shiw people continuing to score. I saw 2 people with scores over 70000. What’s wrong? Do I have a bad version?

11. It’s a simple game (doge2048). It is also addictive. (It’s been clinically proven to be second only to crystal meth… Now that isn’t true of course; my friends from the humanities gave it up quite easily*. I kid, I kid.) Go ahead, give it a go.