nim -- from wolfram mathworld.htm

Sprague-Grundy Value(博弈论)

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            <DIV class=navbar><A 
            href="http://mathworld.wolfram.com/topics/RecreationalMathematics.html">Recreational 
            Mathematics</A>&nbsp;&gt;&nbsp;<A 
            href="http://mathworld.wolfram.com/topics/Games.html">Games</A>&nbsp;&gt;&nbsp;<A 
            href="http://mathworld.wolfram.com/topics/GeneralGames.html">General 
            Games</A>&nbsp;<BR></DIV><BR>
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                <TD class=title vAlign=baseline><SPAN 
                class=nowrap>Nim</SPAN></TD></TR>
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          <TD width="100%"><A href="http://mathworld.wolfram.com/contact/" 
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            <DIV align=center><IMG height=146 alt=Marienbad 
            src="Nim -- from Wolfram MathWorld.files/Marienbad_1000.gif" 
            width=326> </DIV>
            <P class=Text>A game, also called <A class=Hyperlink 
            href="http://mathworld.wolfram.com/Tactix.html">tactix</A>, which is 
            played by the following rules. Given one or more piles (<A 
            class=Hyperlink 
            href="http://mathworld.wolfram.com/Nim-Heap.html">nim-heaps</A>), 
            players alternate by taking all or some of the counters in a single 
            heap. The player taking the last counter or stack of counters is the 
            winner. Nim-like games are also called <A class=Hyperlink 
            href="http://mathworld.wolfram.com/Take-AwayGame.html">take-away 
            games</A> and <A class=Hyperlink 
            href="http://mathworld.wolfram.com/DisjunctiveGame.html">disjunctive 
            games</A>. If optimal strategies are used, the winner can be 
            determined from any intermediate position by its associated <A 
            class=Hyperlink 
            href="http://mathworld.wolfram.com/Nim-Value.html">nim-value</A>. 
            The <A class=Hyperlink 
            href="http://mathworld.wolfram.com/Nim-Heap.html">nim-heap</A> 
            illustrated above corresponds to the game of <A class=Hyperlink 
            href="http://mathworld.wolfram.com/Marienbad.html">Marienbad</A>. 
            </P>
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            <P class=CrossRefs><SPAN class=crosslinkheader>SEE ALSO:</SPAN> <A 
            class=Hyperlink 
            href="http://mathworld.wolfram.com/Marienbad.html">Marienbad</A>, <A 
            class=Hyperlink 
            href="http://mathworld.wolfram.com/Misere-FormGame.html">Mis鑢e-Form 
            Game</A>, <A class=Hyperlink 
            href="http://mathworld.wolfram.com/Nim-Value.html">Nim-Value</A>, <A 
            class=Hyperlink 
            href="http://mathworld.wolfram.com/WythoffsGame.html">Wythoff's 
            Game</A>. <INPUT type=hidden 
            value=http://mathworld.wolfram.com/Nim.html name=as_lq><A 
            href="javascript:document.forms.SearchLinks.submit();">[Pages&nbsp;Linking&nbsp;Here]</A> 
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            <P><IMG height=3 alt="" 
            src="Nim -- from Wolfram MathWorld.files/underline.gif" width=300> 
            </P><SPAN class=crosslinkheader>REFERENCES:</SPAN> 
            <P class=Reference>Ball, W. W. R. and Coxeter, H. S. M. <I><A 
            class=Hyperlink 
            href="http://www.amazon.com/exec/obidos/ASIN/0486253570/ref=nosim/weisstein-20">Mathematical 
            Recreations and Essays, 13th ed.</A></I> New York: Dover, pp. 36-38, 
            1987. </P>
            <P class=Reference>Bewersdorff, J. "Nim: The Easy Winner!" Ch. 21 in 
            <I><A class=Hyperlink 
            href="http://www.amazon.com/exec/obidos/ASIN/1568812108/ref=nosim/weisstein-20">Luck, 
            Logic, &amp; White Lies: The Mathematics of Games.</A></I> 
            Wellesley, MA: A. K. Peters, pp. 169-173, 2005. </P>
            <P class=Reference>Bogomolny, A. "The Game of Nim." <A 
            class=Hyperlink 
            href="http://www.cut-the-knot.org/nim_st.shtml"><TT>http://www.cut-the-knot.org/nim_st.shtml</TT></A>. 
            </P>
            <P class=Reference>Bouton, C. L. "Nim, A Game with a Complete 
            Mathematical Theory." <I>Ann. Math. Princeton</I> <B>3</B>, 35-39, 
            1901-1902. </P>
            <P class=Reference>Gardner, M. "Mathematical Games: Concerning the 
            Game of Nim and Its Mathematical Analysis." <I>Sci. Amer.</I> 
            <B>198</B>, 104-111, Feb. 1958. </P>
            <P class=Reference>Gardner, M. "Nim and Hackenbush." Ch. 14 in <I><A 
            class=Hyperlink 
            href="http://www.amazon.com/exec/obidos/ASIN/0716715899/ref=nosim/weisstein-20">Wheels, 
            Life, and other Mathematical Amusements.</A></I> New York: W. H. 
            Freeman, pp. 142-151, 1983. </P>
            <P class=Reference>Hardy, G. H. and Wright, E. M. <I><A 
            class=Hyperlink 
            href="http://www.amazon.com/exec/obidos/ASIN/0198531710/ref=nosim/weisstein-20">An 
            Introduction to the Theory of Numbers, 5th ed.</A></I> Oxford, 
            England: Oxford University Press, pp. 117-120, 1990. </P>
            <P class=Reference>Kraitchik, M. "Nim."