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Pairs of primes (Posted on 2008-09-21) Difficulty: 2 of 5
A total of 12 digits are placed in a sequence such that the 11 pairs of subsequent digits form 11 different prime numbers.

What is the second digit in this sequence, and what is the last?

  Submitted by pcbouhid    
Rating: 5.0000 (1 votes)
Solution: (Hide)
Dej Mar´s analytic solution is correct. All the possibilities can be found in Charlie´s comment.

For all but the first prime of the 11 in sequence, the first digit of the prime is the second digit of another prime. Since all primes > 5 end in one of the digits 1, 3, 7, 9, this means all but the first prime in the list must begin with one of these digits.

There are 10 two-digit primes beginning with one of these digits: 11, 13, 17, 19, 31, 37, 71, 73, 79, 97. This accounts for 10 of the 11 primes needed. The 11th prime must lead off the sequence, so it can begin with a different number (in fact, the condition that the 2nd digit of the sequence is 1, requires that the first prime is 41 or 61 – the only two digit primes not in the list above to end in 1).

Now, drop the first digit of the full sequence to look at the sequences that can be built only with the primes in the above list. In that list two primes end in 9, but only one begins with 9. The only way this can work is if one of the two primes ending in 9 is the last prime in the sequence. So, final digit has to be 9.

Similarly there are 4 primes that begin with 1, but only three that finish in 1. The only way this can work is if one of the 4 primes beginning in 1 is first in the shortened sequence. So the first digit in the shortened sequence, which is the second digit of the full sequence, must be 1.
There are several solutions, depending on how you arrange the 10 primes, and whether you start off with 41 or 61, but all of them must meet these two conditions.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Puzzle ThoughtsK Sengupta2024-02-20 07:40:30
SolutionEach of the 192 such sequences ...Charlie2008-09-21 13:29:06
SolutionSolutionDej Mar2008-09-21 12:15:16
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