Lewis Carroll's famous game of Doublets is well known. In it you are asked to transform a given word into another by changing only one letter at a time, forming a genuine new word (not a proper name) with each letter change.
Doublets with primes is identical except that instead of playing with words you play with prime numbers, say two 3-digit primes.
Question 1. Can any 3-digit prime be transformed into any other 3-digit prime number following the Doublet rule?
Question 2. What is the longest distance (i.e. the largest number of links required) between two 3-digit primes?
One could ask the same questions about 4-digit primes.
For the first question the answer is yes, for the second question the answer is 6,
To solve the question i used both (Java and Wolfram), the idea is this i made a graph with nodes being the primes with 3-digits and there is a line between two nodes iff the primes representing the nodes are 1-Doublet(meaning with one digit change we can transfer one into another) and then we can state you question as graph theory question which are :
1) is the graph connected ?
2) what is the graph diameter ?
building the graph using Java and answering the questions using Wolfram we are done.
it seems that this is true for any number of digits primes, but i don't think there is a simple proof.