Science of Apps! - Part 3: 2048

Probability Level 4

$2048$ is an insanely addictive game. The objective is simple, bring the same powers of two together to add them together, and keep increasing the powers of $2$ until you reach $2048$ and beyond.

The game starts with $2$ and $4$ tiles, popping up in different places of the $4\times 4$ grid.

To get to the $2048$ tile requires finesse, and a slight bit of luck. In the best scenario, what is the least number of moves required to get the $2048$ tile?

This problem is part of the Science of Apps! series

The answer is 520.

1 solution

Pratham Pandey
May 20, 2014

Informal idea: Only 4's arrive, so 2048/4=512 and since you start with a block out, its actually 512−1=511 have to arrive and the last one needs to be combined $\log _{ 2 }{ \frac { 2048 }{ 512 } }$ i.e. 9 times to make the final 2048 block so a minimum of 520 moves are required, if a perfect game comes up and its perfectly set up when that last 4 comes out, so yes, 520 is the validated absolute minimum.

Whoa! What a coincidence, after solving the question, my rating went up to 2048! :)

Shaan Vaidya - 7 years ago

I had previously seen the question on MathStackExchange and the solution is exactly the same as this. :P

Shaan Vaidya - 7 years ago

Who knows if he got it from here or vice versa

Eric Hernandez - 6 years, 10 months ago

Which is all great, but you actually start with 2 blocks out, not one, making the answer 512-2=510, and the 9 moves for the final 2048 block gives 519 moves, which is actually the ABSOLUTE minimum in the actual game.

Fadil Žilić - 7 years ago

why 9 times?

Gunjas Singh - 7 years ago

I think the answer should be 511 ... Considering the best case scenario(only 4s arrive at perfect places)... To get a 8 u need only 1 move... Therefore to get 16, u need to make two 8's and a final move between them to complete 16... Proceeding in this way------

16-> 1+1+1= 3 32-> 3+3+1= 7 64-> 7+7+1= 15 128-> 15+15+1= 31 256-> 31+31+1= 63 512-> 63+63+1= 127 1024-> 127+127+1= 255 2048-> 255+255+1= 511 ...

Ahnaf Sakib - 7 years ago

i got 511 either

Krit Phuengphan - 7 years ago

The reason you need more than 511 is that you cannot have a 2048 tile by itself + the tile that just spawned when you put the 2 1024s together. Proof: every single move you do backwards, a 2 or 4 must despawn. Go back to the position with 2 1024 tiles and now what was the last move? Using this logic, we end up seeing that we actually need 9 more blocks to come out.

Mark Kong - 7 years ago

Science of Apps! - Part 3: 2048

The answer is 520.

1 solution

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