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?
Let us denote the digits of the set [0,2,4,5,6,8) by b and letter a will
denote the odd digits ,digit 5 excluded. Assuming that A(Alice or Ady) will always use
"a" letters for U and W-
B(Bob or Badguy) should use only "b" letters , otherwise he will
be left only with losing choices.
Lemma 1. ALMOST ALL NUMBERS of
pattern 1b3b7b9 are composite, only 4 samples are prime, but can be avoided by
not playing Y=7.
123479 is prime but
123497 is not <o:p></o:p>
143879 is prime but 143897
153079 is prime but 153097
183479 is prime but neither
183457 nor 183459 are.<o:p></o:p>
A will always play U=1 W=3 AND MOSTLY Y=7.
IFF V,X= 2,4 OR 4,8 OR 5, 0
THEN A PLAYS Y=9
IFF V,X= 8,4 THEN A PLAYS Y=5.
Dull, conservative , but winning.
THERE MIGHT BE SOME SLIPS, feel
free to check and comment.