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.

Facts:  
  123479   is prime but  123497  is not  <o:p></o:p>

143879   is prime but  143897  is not<o:p></o:p>

153079   is prime but  153097  is not<o:p></o:p>

183479   is prime but   neither 183457  nor 183459 are.<o:p></o:p>

STRATEGY:

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.

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Posted by Ady TZIDON on 2013-05-28 12:02:00