### Brain Teasers

# Prime Directive

Arrange the numerals "11337799" to form an 8-digit number (referred to as "ABCDEFGH") that meets the following conditions:

1) ABCDEFGH is a prime number

2) HGFEDCBA (reversing the digits) is a prime number

3) all consecutive 2-digit subsets of ABCDEFGH are prime numbers (AB, BC, CD, DE, EF, FG, GH are all prime)

4) all consecutive 2-digit subsets of HGFEDCBA are prime numbers (HG, GF, FE, ED, DC, CB, BA are all prime)

1) ABCDEFGH is a prime number

2) HGFEDCBA (reversing the digits) is a prime number

3) all consecutive 2-digit subsets of ABCDEFGH are prime numbers (AB, BC, CD, DE, EF, FG, GH are all prime)

4) all consecutive 2-digit subsets of HGFEDCBA are prime numbers (HG, GF, FE, ED, DC, CB, BA are all prime)

### Answer

97131379 (97, 71, 13, 31, 13, 37, 79) and its reverse 97313179 (97, 73, 31, 13, 31, 17, 79)The only available 2-digit prime numbers are: 11, 13, 17, 19, 31, 37, 71, 73, 79, and 97. The number pairs that are prime in both directions are: 11/11, 13/31, 17/71, 37/73, and 79/97 (19 is eliminated since its reverse, 91, is not prime). With these numbers, it can be seen that 9 can only be paired with 7. Therefore, the solution must be of the form 97????79, 9797????, or ????7979. ???? must therefore be 1313, 3131, or 3113. This leaves the possible solution pairs of 97131379/97313179, 97971313/31317979, 97973131/13137979, 97973113/31137979, and 97311379/97311379 (the last pair being a palindrome). Of these, only 97131379 and its reverse, 97313179, are prime numbers.

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