I have a problem with calculating the nimbers for this game. I know that for Kayles the following is true:\begin{array}{c|c} \hline Hight & Nimber \\ \hline 12 & ^*4 \\ \hline 11 & ^*6 \\ \hline 10 & ^*2 \\ \hline 9 & ^*4 \\ \hline 8 & ^*1 \\ \hline 7 & ^*2 \\ \hline 6 & ^*3 \\ \hline 5 & ^*4 \\ \hline 4 & ^*1 \\ \hline 3 & ^*3 \\ \hline 2 & ^*2 \\ \hline 1 & ^*1 \\ \hline 0 & ^*0 \\ \hline \end{array}
My attempt to get to these numbers:
n(0) = mex{} = 0
n(1) = mex{0} = 1
n(2) = mex{0, 1} = 2
However,
n(3) = mex{1, 2} = 0
which is obviously the wrong result.
What I am trying to figure out is how to calculate those nimber values. Any help is appreciated.
$\endgroup$ 21 Answer
$\begingroup$Community wiki answer so the question can be marked as answered:
A third option for the game of length $3$, in addition to leaving a game of length $1$ or a game of length $2$, is to leave two games of length $1$, with value $1\oplus1=0$, so $n(3)=\mathrm{mex}\{0,1,2\}=3$.
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