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    <p>Given a sequence <code>w</code> of distinct numbers, let <code>N(w)</code> be the length of <code>w</code> and let <code>L(w)</code> be the length of the longest increasing subsequence in <code>w</code>. For example, if</p> <pre><code>w = 3 5 8 1 4 </code></pre> <p>then <code>N(w) = 5</code> and <code>L(w) = 3</code>.</p> <p>The game ends when <code>L(w) = N(w)</code>, or, equivalently, <code>N(w) - L(w) = 0</code>. </p> <p>Working the game backwards, if on RobotX's turn <code>N(w) - L(w) = 1</code>, then the optimal play is to remove the unique letter not in a longest increasing subsequence, thereby winning the game. </p> <p>For example, if <code>w = 1 7 3</code>, then <code>N(w) = 3</code> and <code>L(w) = 2</code> with a longest increasing subsequence being <code>1 3</code>. Removing the <code>7</code> results in an increasing sequence, ensuring that the player who removed the <code>7</code> wins.</p> <p>Going back to the previous example, <code>w = 3 5 8 1 4</code>, if either <code>1</code> or <code>4</code> is removed, then for the resulting permutation <code>u</code> we have <code>N(u) - L(u) = 1</code>, so the player who removed the <code>1</code> or <code>4</code> will certainly lose to a competent opponent. However, any other play results in a victory since it forces the next player to move to a losing position. Here, the optimal play is to remove any of <code>3</code>, <code>5</code>, or <code>8</code>, after which <code>N(u) - L(u) = 2</code>, but after the next move <code>N(v) - L(v) = 1</code>. </p> <p>Further analysis along these lines should lead to an optimal strategy for either player.</p> <hr> <p>The nearest mathematical game that I do know is the Monotone Sequence Game. In a monotonic sequence game, two players alternately choose elements of a sequence from some fixed ordered set (e.g. <code>1,2,...,N</code>). The game ends when the resulting sequence contains either an ascending subsequence of length <code>A</code> or a descending one of length <code>D</code>. This game has its origins with a theorem of Erdos and Szekeres, and a nice exposition can be found in <a href="http://www.math.msu.edu/~sagan/Papers/Old/msg.pdf" rel="nofollow">MONOTONIC SEQUENCE GAMES</a>, and this <a href="http://www.mth.msu.edu/~sagan/Slides/msg.pdf" rel="nofollow">slide presentation</a> by Bruce Sagan is also a good reference.</p> <p>If you want to know more about game theory in general, or these sorts of games in particular, then I strong recommend Winning Ways for Your Mathematical Plays by Berlekamp, Conway and Guy. Volume 3, I believe, addresses these sorts of games.</p>
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    1. POAlgorithm for "pick the number up game"
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    1. COthanks for your answer
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