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.

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.

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.

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!!!

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:

- Here’s a tough little 3D version.
- But I actually prefer this tight 2x2x2x2, 4D version.
- Here’s a clever little spin. Combine fractions to add up to 1.
- If you need more room, try playing 2147483648.
- One more silly one, here’s Jenny Peng’s easy version
- Someone figured out how to do the three tile version. Check out 243.
- Many more on this big long list of variations.

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!

A three tile merge game exists here: http://hgentry.github.io/81/

Thanks for linking back to my blog! xD

Fantastic!! Thanks, I’ll add it to the post!

The author of the AI version responded to a stackoverflow question about optimal play — check out his thought process and explanation: http://stackoverflow.com/questions/22342854/what-is-the-optimal-algorithm-for-the-game-2048

Oh yeaaaahhhh. I saw that during my research for the post, but I forgot to include it. Thanks, Andrew!

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There is one with nicely colored fruits for android: https://play.google.com/store/apps/details?id=com.KuuttiEntertainment.Fruits2048

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/

This game just takes all my time!!! but though there are so many apps out there, the only one with the smooth gameplay, no bugs and good sound is the original 2048+

https://itunes.apple.com/us/app/2048+/id841272553?ls=1&mt=8

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.

2048 modified so the 2s and 4s appear in the worst place possible:

http://sztupy.github.io/2048-Hard/

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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!!

That’s really interesting, Navi. Have you used that more than once? I want to try it now.

Okay, I finally finished my own epic post on the subject (and other related stuff). It’s here. I hope you enjoy it.

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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.

https://www.dropbox.com/sh/juuxak8t2twnh8a/elGSs48a7t

Tried to make a portable version in Java. Only of the fibonacci series. :-)

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

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. :)

I made a version of 2048 on a triangular grid. Feel free to add it to your list of variations above :) Tringular 2048: http://creativesparkstudios.com/triangular-2048/

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.

These are really cool! I’m getting some ideas for my math project from here! Thanks!

There’s already a 3072 version of the game where you have to merge threes in the beginning – 3 and 3 to get 6, then 6 and 6 to get 12 and so on :) I’ve played 2048 when it came out and immediately got addicted, and now it’s the same with 3072 because for me personally this game it’s even harder to play! If you want to try it check it out here: https://play.google.com/store/apps/details?id=best.game3072

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?

Interesting.

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:

https://play.google.com/store/apps/details?id=com.sandbox2048

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

Good stuff, Richard. Thanks!