title: Find an algorithm to balance this game
body: <p>I've tried to extract just the part of the problem where I'm having trouble, this is part of a bigger project that I'm doing (not homework). I think it's easiest to describe it as a game (since I need it to finish a game I'm working on). This is a one player game the way I'm describing it.</p>
  
  <p><strong>START</strong> You start with one of two positions:</p>
  
  <ul>
  <li>weight <code>(2,0,-1)</code> and one <code>red</code> play</li>
  <li>weight <code>(3,1,-1)</code> and two <code>red</code> plays</li>
  </ul>
  
  <p><strong>END</strong> The game ends when you have no more plays. The goal is to end the game with weight <code>(0,0,0)</code>.</p>
  
  <p>There are two types of plays <code>red</code> and <code>blue</code>. Given a play, you choose one of four pieces: <code>A,B,C,D</code> that give you additional weight and possibly additional plays. Here are the rules:</p>
  
  <ul>
  <li><p>On a <code>red</code> play:</p>
  
  <ul>
  <li>A adds weight <code>(0,-2,-1)</code></li>
  <li>B adds weight <code>(1,-1,-1)</code> and adds one <code>red</code> play</li>
  <li>C adds weight <code>(2,0,-1)</code> and two <code>red</code> plays</li>
  <li>D adds weight <code>(0,2,1)</code> and two <code>blue</code> plays</li>
  </ul></li>
  </ul>
  
  <p>The rules for <code>blue</code> plays are similar but the first two columns for weights are swapped and the last column is reverse and the types of plays are reversed, so you get this instead:</p>
  
  <ul>
  <li><p>On a <code>blue</code> play:</p>
  
  <ul>
  <li>A adds weight <code>(-2,0,1)</code></li>
  <li>B adds weight <code>(-1,1,1)</code> and adds one <code>blue</code> play</li>
  <li>C adds weight <code>(0,2,1)</code> and two <code>blue</code> plays</li>
  <li>D adds weight <code>(2,0,-1)</code> and two <code>red</code> plays</li>
  </ul></li>
  </ul>
  
  <p><strong>QUESTION</strong> Can this game be won?</p>
  
  <p>I'm trying to write a program that wins the game by choosing plays so that the final balance is <code>(0,0,0)</code> when you have no more plays. Only I can't seem to do it. So now I think that maybe there is no algorithm to win the game. I'd really like to know why that is. If anyone has any ideas how I can prove that, then please let me know!! Or, if you try it and find a way to win, then please tell me that, too. Thanks!!</p>