2048 and algebra (Part 1)

In July and early August of this year, I finally defeated the wildly addicting 2048 game. That’s not to say that I reached the 2048-tile. No, I really defeated the game by reaching the event horizon that literally cannot be surpassed. (This is the usual way I overcome video-game addiction… play the game so much that I get sick of it.)

Over the four weeks or so that it took me to reach the event horizon, I thought of some interesting questions: From looking at only the above screenshot, can I figure out how many moves were needed to reach the final board? Can I calculate how many new 2-tiles and 4-tiles were introduced to the board throughout the course of this game?

It turns out that these questions can be solved with simple algebra. Indeed, if posed in the correct fashion, these questions can be answered using only elementary-school arithmetic. I will discuss the answers to these questions in this series of posts.

It should be noted that the above game board was accomplished in practice mode, and I needed perhaps a couple thousand undos to offset the bad luck of a tile randomly appearing in an unneeded place. I estimate the odds of a skilled player reaching the event horizon in game mode to be about $10^{5000}$ to one. Later in this series, I’ll give my rationale for this estimate.

For what it’s worth, my personal best in game mode was reaching the 8192-tile. I’m convinced that, even with the random placements of the new 2-tiles and 4-tiles, the skilled player can reach the 2048-tile nearly every time and should reach the 4096-tile most of the time.  However, reaching the 8192-tile requires more luck than skill, and reaching the 16384-tile requires an extraordinary amount of luck.

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