Problem 881

3. Let's represent each situation encountered in our game by considering the number of cigarettes in the two corresponding piles as the coordinates $x$ and $y$ of a point on the plane. This will result in a diagram of the following type:

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When playing, we move either parallel to one of the coordinate axes or parallel to the bisector of the (first) coordinate angle, moving closer to the origin $S_{0}$; the player who reaches $S_{0}$ wins. Consequently, the player who reaches one of the two coordinate axes or the bisector loses. Therefore, the player who reaches the point $S_{1}(1,2)$ or $S_{1}^{\prime}(2,1)$ wins, because leaving $S_{1}$ or $S_{1}^{\prime}$ can only result in landing on one of the three named lines. Hence, the player who lands on one of the three lines leading to each of these points loses. Therefore, the player who reaches the point $S_{2}(3,5)$ or $S_{2}^{\prime}(5,3)$ wins, and so on.

Step by step, we arrive at a sequence of winning positions (along which one should move without leaving it throughout the entire game):

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and to a sequence of positions symmetric to the given one relative to the bisector of the coordinate angle.

Note that the difference between the coordinates is equal to the number of the corresponding point. Also note that if we denote by $\alpha$ the famous golden ratio ${ }^{1}$: the root of the equation $\alpha^{2}=\alpha+1$, equal to $(1+\sqrt{5}) / 2$, or approximately 1.618, then the coordinates of the winning point with number $n$ are ${ }^{2}$:

a) the integer part of $(n \alpha)$,

b) the integer part of $(n \alpha) + n$.