99 lines
6.3 KiB
HTML
99 lines
6.3 KiB
HTML
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<h2>Hopping Game</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 391</h3><span style="width:300px;color:#666;">Published on Saturday, 30th June 2012, 11:00 pm; Solved by 188</span></div>
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Let <var>s<sub>k</sub></var> be the number of 1’s when writing the numbers from 0 to <var>k</var> in binary.<br />
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For example, writing 0 to 5 in binary, we have 0, 1, 10, 11, 100, 101. There are seven 1’s, so <var>s</var><sub>5</sub> = 7.<br />
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The sequence S = {<var>s<sub>k</sub></var> : <var>k</var> <img src='images/symbol_ge.gif' width='10' height='12' alt='≥' border='0' style='vertical-align:middle;' /> 0} starts {0, 1, 2, 4, 5, 7, 9, 12, ...}.<br />
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</p>
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<p>
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A game is played by two players. Before the game starts, a number <var>n</var> is chosen. A counter <var>c</var> starts at 0. At each turn, the player chooses a number from 1 to <var>n</var> (inclusive) and increases <var>c</var> by that number. The resulting value of <var>c</var> must be a member of S. If there are no more valid moves, the player loses.
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</p>
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<p>
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For example:<br />
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Let <var>n</var> = 5. <var>c</var> starts at 0.<br />
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Player 1 chooses 4, so <var>c</var> becomes 0 + 4 = 4.<br />
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Player 2 chooses 5, so <var>c</var> becomes 4 + 5 = 9.<br />
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Player 1 chooses 3, so <var>c</var> becomes 9 + 3 = 12.<br />
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etc.<br />
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Note that <var>c</var> must always belong to S, and each player can increase <var>c</var> by at most <var>n</var>.
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</p>
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<p>
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Let M(<var>n</var>) be the highest number the first player can choose at her first turn to force a win, and M(<var>n</var>) = 0 if there is no such move. For example, M(2) = 2, M(7) = 1 and M(20) = 4.
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</p>
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<p>
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Given Σ(M(<var>n</var>))<sup>3</sup> = 8150 for 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>n</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 20.
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</p>
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<p>
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Find Σ(M(<var>n</var>))<sup>3</sup> for 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>n</var> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 1000.
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