How does one get so good at problem solving and pattern recognition that they're able to come up with the official solution to problem C in the google code jam?

Here's the solution

https://code.google.com/codejam/contest/6254486/dashboard#s=a&a=2

It's very interesting. I would never have been able to figure out that "his suggests that b+1 is always a divisor" and then that "Note that b+1 in base b is 11."

I'm trying to think how could I have gone about this in way where I could have stumbled upon the answers.

I guess I could have started at looking at some jam coins and checking what they have in common. Since prime numbers are involved, checking the factors could be a next step. But the b+1 rule, what guide of problem solving principle could navigate into looking into that?

Same with "Note that b+1 in base b is 11." I feel that the mental incentive into looking towards those patterns are not there unless you were aware of both of them (in the context of looking for a solution to this problem) sorta like a catch 22.