The popular game was created by programmer Gabriele Cirulli in one weekend. You can play it online here .
Here are the rules of the game (slightly different from the official rules):
What is the maximum possible score that one can obtain using the above rules of
Bonus: What is the maximum attainable value for a tile using the official rules of
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Let us show that 2 1 7 is the maximum possible number that can be obtained.
First, one can easily show that 2 1 8 is impossible. This is because constructing 2 1 8 requires 2 tiles with the value 2 1 7 . Now, suppose that we already have one of the 2 1 7 tiles and are now trying create the second one. That one would require 2 tiles with the value 2 1 6 and so on. Ultimately, we find that we need 2 tiles with the value 4 , but adding up all of the tiles we already have gives us 1 7 tiles in total, which is impossible on a 4 by 4 board, as seen in the image below.
Note that 2 1 8 = 2 6 2 1 4 4 is impossible. Created with the online game programmed by its creater. It took a couple of hours playing the game until it resulted in this situation (just kidding).
Second, one can easily show that 2 1 7 is possible to construct. One way uses a snake-like strategy, gradually increasing the value of a single tile in a corner, with the new tiles appearing only if convenient and always in a convenient location, with the final result shown in the above picture. The exact details are left as an exercise to the reader. I will post one if I have time.
Ok, let's show that 2 1 7 is possible to construct. For this strategy, we just need to have only 4 's appearing. Let f ( x , y ) denote a 4 appearing at the location of a square ( x , y ) on the board, with ( 0 , 0 ) denoting the upper left corner, ( 3 , 0 ) the upper right, ( 0 , 3 ) the lower left, ( 3 , 3 ) the lower right, and so on.
Then, follow the below rules:
Repeat the above pattern until the squares are, from upper left to bottom left, 4, 8, 16, 32. Then, do the following:
Yes, we're slowly getting there. Now, you probably see how the building pattern works. Repeat this until we have, from the bottom left square to upper left square, 64, 32, 16, 8. Then,
Yep, you're probably realizing that this process is similar to the one before. We're slowly building such that the lower left corner has the largest number. However, instead of swiping down as in the first column, we're swiping up for numbers in the second column. Continue this for a certain time, but then you swipe down for the third column and up for the fourth column (a snake-like pattern). Finally, we'll get the result in the picture above, if continued until no spaces are left.