Alice and Bob play a game. Starting with Alice, they alternate in selecting digits for a 6-digit decimal number UVWXYZ that they construct from left to right. Alice chooses U, then Bob chooses V, then Alice chooses W, and so on. No digit can be repeated. Alice wins if UVWXYZ is not a prime. Can Alice always win?

(In reply to

An approach, no guarantee by xdog)

The answer is YES if you find U and W s.t. for any UVWX Alice used two of the 1,3,7,9 and can chose Y such that when Bob adds the last odd digit it is a composite number.

Hint:** think mod 3 first.**