Podliiskyi O.K.

In the middle cell of a strip $1 \times 2005$, a chip is placed. Two players take turns moving it: the first player moves the chip one cell in any direction, then the second moves it two cells, the first - four cells, the second - eight cells, and so on (the $k$-th move is made by $2^{k-1}$ cells). The player who cannot make the next move loses. Who can win regardless of the opponent's play?
Note that the first player makes moves of length $1, 4, 1024$. We will show that the first player will be able to make a move of length 1024 (then the second player will lose, as the second player would have to make a move of length 2048, which is impossible).

Note that $1+2+4+\ldots+128 < 1024$, so the first player will be able to make a move of length 1024 (towards the central cell).

## Answer

First.

Submit a comment