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POFind an algorithm to balance this game
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copied!<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>

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