title: In the game 2048, what is the biggest theoretical tile?
body: <p>In the game <a href="http://gabrielecirulli.github.io/2048/">2048</a> what is the biggest tile that can be achieved, assuming a player playing optimally and tile spawning at the most optimal place?</p>
  
  <p>Naively I would say that the biggest achievable tile is <code>65536 * 2 = 131072</code> because it seems that the best possible board is the following:</p>
  
  <pre><code>  4     4     8     16
   256   128    64    32
   512   1024  2048  4096
  65536 32768 16384  8192
  </code></pre>
  
  <p>But I'm not sure if </p>
  
  <ol>
  <li>it's correct</li>
  <li>how to prove that my intuition is indeed correct.</li>
  </ol>
  
  <p>(sorry if I should have asked on <a href="https://gaming.stackexchange.com/questions/ask">gaming.stackexchange</a>, but this is more of a CS question than a gaming one afaict)</p>