OMCC 1999 Problem 3

The digits of a calculator (with the exception of 0) are shown in the form indicated by the figure below, where there is also a button +": ![Image](6965) Two players $A$ and $B$ play in the following manner: $A$ turns on the calculator and presses a digit, and then presses the button +". $A$ passes the calculator to $B$ , which presses a digit in the same row or column with the one pressed by $A$ that is not the same as the last one pressed by $A$ ; and then presses + and returns the calculator to $A$ , repeating the operation in this manner successively. The first player that reaches or exceeds the sum of 31 loses the game. Which of the two players have a winning strategy and what is it?