I am required to give an example of nim game such that:

  1. the initial position is an N-position,
  2. there are exactly three optimal moves from the initial position,
  3. every pile has at least 20 chips,
  4. every pile has a different (nonempty) number of chips

How could I design such a nim? I am in a mess now. Could anyone help me? If possible, could you teach me how to think in solving this problem? I'm really messed...

  • $\begingroup$ Do you know how to decide if a position is N? And what a winning move must do? $\endgroup$ Jun 28 '18 at 5:57
  • $\begingroup$ @HagenvonEitzen I know Bouton's Thm. But I don't know how to use it. $\endgroup$
    – user450201
    Jun 28 '18 at 7:12

We want three different positive integers $a,b,c\ge20$ such that $d:=a\oplus b\oplus c$ is non-zero (the criterion for $N$-positions) and all three piles can be decreased in a way to make this sum zero, i.e., $a\oplus d<a$, $b\oplus d<b$, and $c\oplus d<c$. For the last conditions, it suffices to achieve that the most significant bit of $d$ is set in all of $a,b,c$. So fore example if all of $a,b,c$ are between $16$ and $31$, inclusive, then so will be $d$. In fact, if we try the minimal possible (according solely to $a,b,c\ge20$) choices for $a,b,c$, $$\begin{align}a&=20=10100_2\\b&=21=10101_2\\c&=22=10110_2\end{align}$$ we find $d=10111_2=23$ and $$\begin{align}a\oplus d&=00011_2=3<a\\b\oplus d&=00010_2=2<b\\c\oplus d &=00001_2=1<c\end{align}$$ i.e., the three winning moves are to take $17$ tokens off $a$, or $19$ tokens off $b$, or $21$ tokens off $c$.


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