94 lines
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94 lines
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<h2>Stone Game III</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 366</h3><span style="width:300px;color:#666;">Published on Saturday, 7th January 2012, 07:00 pm; Solved by 326</span></div>
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Two players, Anton and Bernhard, are playing the following game.<BR />
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There is one pile of n stones.<BR />
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The first player may remove any positive number of stones, but not the whole pile.<BR />
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Thereafter, each player may remove at most twice the number of stones his opponent took on the previous move.<BR />
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The player who removes the last stone wins.
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E.g. n=5<BR />
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If the first player takes anything more than one stone the next player will be able to take all remaining stones.<BR />
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If the first player takes one stone, leaving four, his opponent will take also one stone, leaving three stones.<BR />
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The first player cannot take all three because he may take at most 2x1=2 stones. So let's say he takes also one stone, leaving 2. The second player can take the two remaining stones and wins.<BR />
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So 5 is a losing position for the first player.<BR />
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For some winning positions there is more than one possible move for the first player.<BR />
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E.g. when n=17 the first player can remove one or four stones.
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<P>
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Let M(n) be the maximum number of stones the first player can take from a winning position <i>at his first turn</i> and M(n)=0 for any other position.
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<img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' />M(n) for n<img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' />100 is 728.
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<P>
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Find <img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' />M(n) for n<img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' />10<sup>18</sup>.
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Give your answer modulo 10<sup>8</sup>.
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