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Antti H S Laaksonen 2017-02-18 17:13:06 +02:00
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@ -209,7 +209,7 @@ is as follows:
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
Surprisingly enough, in this game, Surprisingly, in this game,
all even-numbered states are winning states, all even-numbered states are winning states,
and all odd-numbered states are losing states. and all odd-numbered states are losing states.
@ -256,7 +256,7 @@ $[10,12,5]$ is $10 \oplus 12 \oplus 5 = 3$,
so the state is a winning state. so the state is a winning state.
But how is the nim sum related to the nim game? But how is the nim sum related to the nim game?
We can explain this by studying how the nim We can explain this by looking at how the nim
sum changes when the nim state changes. sum changes when the nim state changes.
~\\ ~\\
@ -395,7 +395,7 @@ The \key{Grundy number} for a game state is
\[\textrm{mex}(\{g_1,g_2,\ldots,g_n\}),\] \[\textrm{mex}(\{g_1,g_2,\ldots,g_n\}),\]
where $g_1,g_2,\ldots,g_n$ are Grundy numbers for where $g_1,g_2,\ldots,g_n$ are Grundy numbers for
states to which we can move from the state, states to which we can move from the state,
and the mex function returns the smallest and the mex function gives the smallest
nonnegative number that is not in the set. nonnegative number that is not in the set.
For example, $\textrm{mex}(\{0,1,3\})=2$. For example, $\textrm{mex}(\{0,1,3\})=2$.
If there are no possible moves in a state, If there are no possible moves in a state,
@ -787,10 +787,10 @@ For example, when $n=8$, the possibilities
are $1+7$, $2+6$ and $3+5$, so are $1+7$, $2+6$ and $3+5$, so
\[f(8)=\textrm{mex}(\{f(1) \oplus f(7), f(2) \oplus f(6), f(3) \oplus f(5)\}).\] \[f(8)=\textrm{mex}(\{f(1) \oplus f(7), f(2) \oplus f(6), f(3) \oplus f(5)\}).\]
In this game, the value of $f(n)$ is based on values In this game, the value of $f(n)$ is based on the values
of $f(1),\ldots,f(n-1)$. of $f(1),\ldots,f(n-1)$.
The base cases are $f(1)=f(2)=0$, The base cases are $f(1)=f(2)=0$,
because it is not possible to divide heaps because it is not possible to divide the heaps
of 1 and 2 sticks. of 1 and 2 sticks.
The first Grundy numbers are: The first Grundy numbers are:
\[ \[