MathNet 2007-part-01 — 187 problems (EN–VI) Question 20

Alex and Bruno together write a natural number with 6 distinct digits. Each one, on his turn, writes a digit to the right of the last digit written by the other. Alex starts with the first digit on the left and Bruno finishes with the last digit on the right. (It is forbidden to write a digit that has already been used.)
Bruno wins if the 6-digit number is prime. Otherwise, Alex wins.
Determine which of the two players has a winning strategy and explain how he must play to win no matter how well the other plays.