They figure out how to solve the next problem by learning from their experiences of solving the previous problems.
Nope. There is something that AI will never have: Common Sense. AI solves the problems faster, not because it has "experience memories".
Actually AI can and will develop common sense. Forget common sense, AI is capable of "Intuition", a trait that was once thought to be exclusive to humans only. I would suggest you to read about Google's Alpha Go project. You will get all the answers there. :)
It's all about probability and maths. As I said an intelligent computer makes faster calculations.
I checked the project. There are a lot of neurologists involved. Thing is Brain is an amazing organ and mystery. There are a lot of its secrets that humans haven't figured out. Without a complete understanding of brain's functionalities there won't be any algorithm or program for AI.
But Go is a game in which not only logic but intuition is used as well. When some world champions were asked why they made a particular move, they said that it just felt right. The same was done in many cases by Alpha Go, making very unorthodox moves.
Ask the AI if it did a move because of intuition. For real.