For the past few weeks I have been completely addicted to the game of 2048. 2048 has a simple design and concept, but it has had amazing popularity, last week being #1 free app on Apple Appstore. The game involves sliding tiles together, starting with 2s. Each time two identical tiles collide then they join together to form a new one that is twice the value of the previous. So, two 2s slide together to form a 4. Two 4s make an 8…so on and so forth.

This type of doubling is exponential growth and can be modeled by the formula f(x) = 2^{x}.

Exponential growth tends to explode rapidly out of control. The main examples used to describe it is a population of multiplying bunnies, or having your allowance doubled each day for a month starting at a penny (you become an instant millionaire!) Suddenly the 2s and 4s tiles are sliding together to form 8s, 16s, 32s, 64s, 128s, 256s, 512s, 1024s, and 2048s. Experienced players know that the game is not over at 2048, despite being the name of the game. I’ve gone as far as 4096 after an unhealthy and concerning amount of time trying.

Is there a limit to how high the tiles can get? The answer is yes. Given the allotted space on the 4×4 grid the tiles are eventually constricted from doubling any higher. The largest possible tile is 131,072 and is achieved by creating a long string of progressively smaller numbers and randomly receiving a 4 for the final one ( I wonder what color that one would be? ) Take a look at this link to understand just how much work would go into achieving this score : http://imgur.com/Tl7Nldi

131,072 is 2^{17}. 17 is not a “random” number, it is derived from the fact that there are 16 spaces on the board. Technically, this should mean the answer is 2^{16}, but this is not always the case because the game is programmed so that a 2 or 4 could appear in the final cell. Since 4 is 2 squared, it raises the maximum from 2^{16 }to 2^{17}.

From this we can determine what the maximum tile would be for larger game boards, say instead of a 4×4 that it was 5X5. A 5X5 board would have the largest tile be 2^{26 }or 67,108,864, again depending on the last one being a 4 and not a 2. Looking in the other direct, a 2×2 board can only have a maximum tile of 2^{5} or 32. This depends on the last tile being randomly generated as a 4, but it tends to be a 2 so personally I have never made it past 16. This game can be played here: http://2048.mobi/4/

Since 2048 is 2 to the eleventh power, a more politically correct game would only have 10 tiles (perhaps arranged as a 2×5 rectangle). An incredible number of variations to 2048 exist with different board sizes, styles, numbers, etc. For a look at some of the many variations of 2048 and the classic game itself go to this link: http://mathmunch.org/2014/03/24/2048-2584-and-variations-on-a-theme/

So far my high score on a 5×5 is almost 4 mill points with a 131,072 block. my highest on a 4×4 us considerably lower with a score of around 320k and a highest block of 16,384. Thanks for the blog/page it was very informative.

Scratch that lol almost 3 mill points and climbing.