singingbanana's video: 2048 Induction Extra

An extra bit from our 2048 video on Steve Mould's channel. Full video here: 2048 strategy and maths http://youtu.be/OO4tA5i7X9g Play 2048 here http://gabrielecirulli.github.io/2048/ ---------------------------- A little bit on the maximum achievable tile. I said here you need k free cells to achieve 2^k and the maximum achievable tile is therefore 2^16. This was assuming we only generate 2-tiles. I also show a formula to calculate the total score of making the 2^k tile which is (k-1)2^k. That means the maximum score is when you fill the board from 2^16 to 2, which is sum_i=1^16 (i-1)2^i = 1,835,012. If we include generating 4-tiles as well, you can actually go one step further and achieve the 2^17 tile. In that case the maximum score would be when we fill the board from 2^17 to 2^2. If I did that using 4-tiles only then that would double the maximum score from 1,835,012 to 3,670,024. If we use 2-tiles only, with a few exceptions, then I reckon the maximum score is sum_i=2^17 (i-1)2^i - 16*4 = 3,932,100. (You need to subtract 4 sixteen times because I need to generate sixteen 4-tile for free to fill the board from 2^17 to 4).