308. There is a strip of grid paper with 15 cells, and the cells are numbered $0,1,2, \ldots, 14$. Two players take turns moving a token, which is placed on one of the cells, to the left by 1, 2, or 3 spaces. The player who cannot make a move loses. For which initial positions of the token does the first player win, and for which does the second player win?
$\triangle$ Let's consider several cases (Fig. 38).

![](https://cdn.mathpix.com/cropped/2024_05_21_0aa0e6ac168e80891953g-098.jpg?height=111&width=1223&top_left_y=2212&top_left_x=428)

Fig. 38

1) Suppose the chip is initially on the cell numbered 0. Then the second player wins without making a single move.
2) Suppose the chip is initially on one of the cells numbered 1, 2, or 3. Here, the first player wins by moving the chip to cell 0 in one move.
3) Assume the chip is initially on cell 4. Then the second player wins: if the first player moves the chip left by 1, 2, or 3 spaces, the second player should move it back by 3, 2, or 1 spaces, respectively, so that it ends up on cell 0. We will call this response a "complement to 4."
4) Suppose the chip is initially on one of the cells numbered 5, 6, or 7. What should be done here?

The first player should move the chip 1, 2, or 3 spaces, respectively, so that it lands on cell 4, after which the situation reduces to the previous case but with the players' roles reversed. Therefore, the first player wins.

5) Suppose the chip is on cell 8. Here, the second player wins by twice complementing the number of spaces the first player moved the chip to 4.

Further, if the chip is initially on cell 9, 10, or 11, the first player wins (think about how), if on cell 12, the second player wins, and if on cell 13 or 14, the first player wins again.

Answer: the second player wins if and only if the chip is initially on cell 4, 8, or 12.