Prime Me!
I have a number that I like. It is 6 digits long and consist of the digits 1, 2, 3, 4, 5, and 6.
The reason why I like the number is because the sum of any 2 adjacent numbers is a prime number, and the sum of the first and last number is also a prime number!
If the 1st digit of the number is 1, what is the 4th digit of the number?
For example, 123456 is not the number, as the sum of 4 and 5 is 9, which is not prime.
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We may find pairs of { 1 , 2 , 3 , 4 , 5 , 6 } such that their sum is a prime number.
( 1 , 2 ) , ( 1 , 4 ) , ( 1 , 6 ) , ( 2 , 3 ) , ( 2 , 5 ) , ( 3 , 4 ) , ( 5 , 6 )
Order does not matter. Then make a graphs with 6 nodes, named by the numbers { 1 , 2 , 3 , 4 , 5 , 6 } . there would be an edge between two nodes iff their corresponding numbers make a pair that exist in the set above. Making a desired number is equivalent to making a Hamiltonian cycle on the graph. Since the first digit (from the left) is 1 , we should start from node 1 and return to node 1 . node 1 has three neighbours. If we go to node 2 , in the first step, then we are gonna have to take one of the two cycles of length 4 to return to 1 . But, a Hamiltonian cycle is of length 6 . If we go to either 4 or 6 from 1 , then we can successfully complete a Hamiltonian loop and in both cases, the fourth digit is 2 .