Problem 6, 2022

A strip of \(1 \times n\) cells is given, where \(n > 10\) is a natural number, and its cells are numbered \(1, 2, \dots, n\) from left to right. Cell number \(10\) is black and carries a token; every other cell is white. Two players move alternately, as follows.

The player who moves first moves the token to any white cell and colours that cell black. On each later move, the player to move moves the token to a white cell, with the requirement that the token jump over at least one black cell, that is, that at least one black cell lie strictly between the cell it leaves and the cell it reaches; the cell it lands on is then coloured black as well. The player who is first unable to move loses.

Determine, in terms of \(n\), which of the two players has a winning strategy: the one who moves first, or the one who moves second.