Nimbers for Kayles 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.

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 $n(3)=\mathrm{mex}\{0,1,2\}=3$.
