Monthly Archives: March 2014

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.

2048

2048

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!

Making Pi, Transcending Pi, and Cookies

Welcome to this week’s Math Munch– and happy Pi Day!

What does pi look like? The first 10,000 digits of pi, each digit 0 through 9 assigned a different color.

You probably know some pretty cool things about the number pi. Perhaps you know that pi has quite a lot to do with circles. Maybe you know that the decimal expansion for pi goes on and on, forever and ever, without repeating. Maybe you know that it’s very likely that any string of numbers– your birthday, phone number, all the birthdays of everyone you know listed in a row, followed by all their phone numbers, ANYTHING– can be found in the decimal expansion of pi.

But did you know that pi can be approximated by dropping needles on a piece of paper? Well, it can! If you drop a needle again and again on a lined piece of paper, and the needle is the same length as the distance between the lines, the probably that the needle lands on a line is two divided by pi. This experiment is called Buffon’s needle, after the French naturalist Buffon.

If the angle the needle makes with the lines is in the gray area (like the red needle’s angle is), it will cross the line. If the angle isn’t, it won’t. The possible angles trace out a circle. The closer the center of the needle (or center of the circle) is to the line, the larger the gray area– and the higher the probability of the needle hitting the line.

This may seem strange to you– but if you think about how the needle hitting a line has a lot to do with the distance between the middle of the needle and the nearest line and the angle it makes with the lines, maybe you’ll start to think about circles… and then you’ll get a clue about the connection between this experiment and pi. Working out this probability exactly requires some pretty advanced mathematics. (Feeling ambitious? Read about the calculation here.) But, you can get some great experimental results using this Buffon’s needle applet.

Click on the picture to try the applet.

Click on the picture to try the applet.

I had the applet drop 500 needles. Then, the applet used the fact that the probability of the needle hitting a line should be two divided by pi and the probability it measured to calculate an approximation for pi. It got… well, you can see in the picture. Pretty close, right?

Here’s another thing you might not know: pi is a transcendental number. Sounds trippy– but, like some other famous numbers with letter names, like e, pi can never be the solution to an algebraic equation involving whole numbers. That means that no matter what equation you give me– no matter how large the exponent, how many negatives you toss in, how many times you multiply or divide by a whole number– pi will never, ever be a solution. Maybe this doesn’t sound amazing to you. If not, check out this video from Numberphile about transcendental numbers. Numbers like pi and e don’t do mathematical things we’re used to numbers doing… and it’s pretty weird.

Still curious about transcendental numbers? Here’s a page listing the fifteen most famous transcendental numbers. My favorite? Definitely the fifth, Liouville’s number, which has a 1 in each consecutive factorial numbered place.

Escher cookies 1Finally, maybe you don’t like pi. Maybe you like cookies instead. Lucky for you, you can do many mathematical things with cookies, too. Like make cookie tessellations! This mathematical artist and baker made cookie cutters in the shapes of tiles from Escher tessellations and used them to make mathematical cookie puzzles. Beautiful, and certainly delicious.

If you happen to have a 3D printer, you can make your own Escher cookie cutters. Here’s a link to print out the lizard cutter. If you don’t have a 3D printer, you could try printing out a 2D image of an Escher tessellation and tracing a tile onto a sheet of paper. Cut out the tile, roll out your dough, and slice around the outside of the tile to make your cookies. If you do it right, you shouldn’t have to waste any dough…

Here’s hoping you eat some pi or cookies on pi day! Bon appetit!