World Cup

What's the maximum number of points a team can get in the group stage of the World Cup and not advance to the knockout round?

2048

The game 2048, a puzzle game inspired by the game Threes, has taken the internet by storm--Sudoko style. Numerous variations have been spawned, including a 3-D version,

a 4-D version,
 
 a Fibonacci version called 1597 (since 2048 is sadly not a Fibonacci number), 

and more. Math Munch did a nice write-up on a number of different versions a few months ago.

As you can see from the above pictures, I've been playing a lot. You can continue to play after "winning", which is just cruel to those of us with a healthy dose of competitive ire. Then there's the Powers of 1 version where 1's are continuously generated in the fruitless battle to get to 2 to remind you of the parallels between this game and heroin. In an attempt to maybe justify the time I've spent on this game as I'm clearly still in denial, I've been thinking of a few questions related to the puzzle.

Where's the math?

The naive answer to this is that the puzzle involves numbers, and adding (or is it multiplying), but is the version involving cats really any different?

 

Numbers are used as a convenient (and familiar) way to keep track of what turns into what, but after a while you do remember (creepily) that two orange tabbies make a bengal. 

The phrase "problem solving" might also be thrown around, and I agree that a thoughtful strategy will lead to greater success than mashing randomly on the arrow keys. That said, one of the things that I am particularly interested in is naming these problem solving techniques. Can you see the strategy I employ in the following game? What might you call this?

There's also something to be said for being able to visualize how the tiles will move, not to mention how they relate to one another. Knowing that 32 and 128 are separated by "two levels" is helpful. I've noticed in my own playing that I have to think about this more with the Fibonacci version (and the cat version).

Finally, and most importantly, I would argue that the greatest mathematical aspect of this game lies in its inherent flexibility. The basic structure and rules cry out to be changed, and the people have responded. Altering the structure of the 2048 universe is no different than asking how we might make sense of the square root of negative one.

Which version is hardest?

Well, the Powers of One version is the hardest, although 9007199254740992 probably runs a close second.

Between the 2D, 3D, 4D, and Fibonacci version, though, this is a harder question. Anecdotally, I'd arrange them as following, from easiest to hardest: 3D (once you make sense of the structure), Fibonacci, 2D, 4D. I've been thinking about how to measure this in a more rigorous way, though. 

There seem to be differing levels of flexibility in the different versions. If you try and stick to the strategy of keeping your largest number in a corner, there are really only 2 moves you can safely make in the 2D version (south and west), unless the bottom row is complete and then there are three safe moves (south, west, and east). In the 3D version, there are three moves that are always safe.

There are 27 total squares in the 3D version, while the rest are limited to 16. This also seems to allow for greater flexibility in the 3D version.

The Fibonacci version is made easier by the fact that there are a greater number of pairs that combine. An 8, for example, can be combined with either a 5 or a 13. In all the other versions an 8 can only be paired with an 8.

And the most important question of all: where did the last five hours go?