Question 34
The sequence of numbers $1, 2, 3, \ldots, 100$ is written on the blackboard. Between each two consecutive numbers a square box is drawn. Player $A$ starts the game and the players $A$ and $B$ alternate the moves. In each turn a player chooses an empty box and places "+" or "." sign in it. After all the boxes are filled the expression on the blackboard is evaluated and if the result is an odd number the winner is $A$. Otherwise the winner is $B$. Determine which of the players has a winning strategy and what the strategy is.