2048, 2584, and variations on a theme

Welcome to this week’s Math Munch! It’s a week of mathematical games, including a devilish little game and variations on the theme.



First up, check out this simple little game called 2048. Really, you must go try that game before reading on.

Gabriele Cirulli

Gabriele Cirulli

2048 was created by Gabriele Cirulli, a 20-year old who lives in northern Italy. He was inspired by a couple of very similar games called 1024 and threes, and he wanted to see if he could code a game from scratch. Nice work, Gabriele! (Stay tuned for a Q&A with Gabriele. Coming soon.)

The first time I played, I thought randomly moving the pieces around would work as well as anything, but wow was I wrong. Give it a try and see how far you get. Now watch how this AI (artificial intelligence) computer program plays 2048. You’ll probably notice some patterns that will help you play on your own.

A beautiful chain of powers of two.

A beautiful chain of powers of two.  Can you solve from here?

Did you notice that the smallest tiles are 2’s, and you can only combine matching tiles to create their double? This makes all of the tile values powers of two! (e.g. 2048=2^11) These are the place values for the binary number system! (Did you see our recent post binary?) This has something to do with the long chains that are so useful in solving the game. It’s just like this moment in the marble calculator video.

4, a silly, but interesting little variation

4, a silly, but interesting little variation

If you’re finding 2048 a bit too hard, here’s an easier version.  It’s called 4. It’s a little silly, but it’s also quite interesting. After you make the 4 tile (tying the world record for fewest moves), click “keep going” and see how far you can get. I’ve never been able to get past the 16 tile. Can anyone make the 32? What’s the largest possible tile that can be made in the original 2048 game? Amazingly, someone actually made a 16384 tile!!!

2584, the Fibonacci variant of 2048

2584, the Fibonacci version of 2048

Silly versions aside, there are lots and lots of ways you could alter 2048 to make an interesting game. I wondered about a version where three tiles combined instead of two, but I couldn’t quite figure out how it would work. Can you? (See below.) When I thought about different types of numbers that could combine, I thought of the perfect thing. The Fibonacci numbers!!! 1, 1, 2, 3, 5, 8, 13, 21, … The great thing is that someone else had the same idea, and the game already exists! Take some time now to play 2584, the Fibonacci version of 2048.

2048 and 2584 might seem like very similar games at first, (they’re only 536 apart), but there are some really sneaky and important differences. In the Fibonacci version, a tile doesn’t combine with itself. It has two different kinds of tiles it can match with. I think this makes the game a little easier, but the website says 2584 is more difficult than the original. What do you think?

I have a few more 2048 variations to share with you, as if you didn’t have enough already. These are my favorites:

I hope you dig into some of these games this week. Really think and analyze. If you come up with clever strategies or methods to solve these puzzles, please let us know in the comments. Have a great week, and bon appetit!

38 responses »

  1. Pingback: Voldoening (vier observaties over 2048) » Sylvia's blog

  2. I’m pretty sure that in a “combine powers of two” game, you can never make a power of two higher than the number of cells in the grid, because there must come at least one point when you have all those powers of two on the board at the same time. By the way, see also DIV and DIVE: http://alexfink.github.io/dive/

  3. The highest possible tile would be two to the seventeenth since you could have two fours, one of which would have just appeared after the last slide, and then also have the next fourteen powers of two lined up in order all the way up to two to the sixteenth, and then all of them could be combined in order and then get two to the seventeenth. This is an incredibly unlikely scenario, but I believe it’s theoretically possible.

    • Unless of course somebody comes up with a 2048 version of Hatetris in which case that 4 will never show up. Actually, you would never even get close to that far. You might not even be able to get the 2048 tile. That’s an interesting question: If you’re playing an opponent, can a skilled one always prevent you from getting the 2048 tile? Or can a sufficiently skilled play always outplay even the most skilled opponent and get that tile? I guess that’s similar to asking if every game is potentially winnable.

  4. Pingback: Favorited Tweets #1 | Growing Exponentially

  5. I played the fibonacci game and discover something that surprisinly worked!! I beat the game after the fist time (needed some luck of course) and then tried something out. From the very beginning you move as following: right , down, up, left and repeat. I thought it’d go nowhere and looked really messy at some point… All of a sudden the spread out tiles combined, cleared almost everything and built the 377 tile!!

  6. Pingback: The Greatest Tile on 2048 | All the Calculations

  7. Played Fibonacci version and win. It’s easyer than 2048. That it’s pretty esay as well. I always reach 2048. My best has been 8192.

  8. I absolutely love this game, me and all my friends constantly play it. my friend ed beat the game on his second try!

  9. My family is obsessed with the 2048 game. I appreciated this article because even a laymen like me could understand it. If there was something I didn’t know you included a link for further investigation. For example, I had no idea what the Fibonacci sequence was so I simply clicked on the link, which led me to a video. The video made it look like the Fibonacci sequence creates a cochlear spiral thingamajig, which probably has a fancy name. Time for more mathematical investigation….

    • Hey Autumn. I love your comment. The mathematical, link-clicking, investigation is exactly what this site is about. I’m so glad you’ve found it.

      As for the Fibonacci numbers, you can also try to search our site to find more articles that include them. There are probably lots. 🙂

    • Oh, interesting! I imagined something slightly different. For a triangular 2048, I think I would have had three directions instead of 6. In your game they would be the Q, E, and S directions. When you hit Q for example, the triangles slide as far as they can to the left, in the same they slide as far as they can in 2048.

      Maybe you’ll code that version too!?

      • Yes, that would be interesting to explore – maybe a future project! One problem would be that in a triangular grid the tiles have one of two orientations (pointing up or down) so the whole concept of sliding gets obscured a little bit. But maybe it could be done.

      • Great point about the orientation!! We could let let them change orientation to fit their cell. That seems like the simplest way to do things.

    • Jen- it seems like the numbers on the tiles are a little different (50% larger in 3072), but that 3072 and 2048 are actually the exact same underlying game. Aren’t the strategy, objectives, and game play exactly the same?


  10. I made a “sandbox” version of 2048 for Android that allows any grid size from 1×1 to 32×32. You can also change the starting number from 2 to 3 or 4 or nearly anything else you could want. You can even edit the type of tiles inserted after each move and change the probabilities of where they are inserted.

    It’s on Google Play, 2048 Sandbox:

    I’d be delighted if you added it to your list of variants!

  11. Pingback: The Art of Merete Rasmussen, a Game About Squares, and VAX! | Math Munch

  12. This game is really fun and addicting. The only thing I don’t like about is, it takes a lot of thinking, where to move next so everything works out.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s