There are pawns at each of two ending cells of 1 × 20 strip

Question

There are pawns at each of two ending cells of 1 × 20 strip. In a single step a player can shift one of the pawns towards the second one by one or two cells. It is forbidden to jump over a pawn. The player who cannot make a step loses the game. Who has a winning strategy - the player who starts the game or his opponent?

Answer 1

The game is equivalent to a simple takeaway game. The game starts with a pile of $18$ stones. Each player in turn must take either one or two stones from the pile; the player who takes the last stone wins, since the other player cannot make a move.

Observe that if there are $3$ stones left, the player whose turn it is must lose: no matter what that player does, the other player can always take the last stone. Thus, if you can leave your opponent $3$ stones, you’ll win. If you leave your opponent $4$ or $5$ stones, however, your opponent can leave you $3$ stones and win. What if you leave your opponent $6$ stones? $9$ stones?

Answer 2

The losing positions for the player to move are those in which the number of empty squares between the pawns is a multiple of $3$. So the second player wins.