95 lines
6.0 KiB
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95 lines
6.0 KiB
HTML
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<h2>Stone Game II</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 325</h3><span style="width:300px;color:#666;">Published on Saturday, 19th February 2011, 01:00 pm; Solved by 333</span></div>
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A game is played with two piles of stones and two players. At her turn, a player removes a number of stones from the larger pile. The number of stones she removes must be a positive multiple of the number of stones in the smaller pile.
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E.g., let the ordered pair(6,14) describe a configuration with 6 stones in the smaller pile and 14 stones in the larger pile, then the first player can remove 6 or 12 stones from the larger pile.
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The player taking all the stones from a pile wins the game.
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A <i>winning configuration</i> is one where the first player can force a win. For example, (1,5), (2,6) and (3,12) are winning configurations because the first player can immediately remove all stones in the second pile.
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<p>
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A <i>losing configuration</i> is one where the second player can force a win, no matter what the first player does. For example, (2,3) and (3,4) are losing configurations: any legal move leaves a winning configuration for the second player.
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Define S(<var>N</var>) as the sum of (<var>x</var><sub><var>i</var></sub>+<var>y</var><sub><var>i</var></sub>) for all losing configurations (<var>x</var><sub><var>i</var></sub>,<var>y</var><sub><var>i</var></sub>), 0 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>x</var><sub><var>i</var></sub> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>y</var><sub><var>i</var></sub> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <var>N</var>. We can verify that S(10) = 211 and S(10<sup>4</sup>) = 230312207313.
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Find S(10<sup>16</sup>) mod 7<sup>10</sup>.
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