It’s interesting to note that in regular mathematics, induction on the natural numbers is axiomatic.[1] That is, you can’t get there from within the framework you have built up to then so you need to build an extra axiom into the system to make it work. Both the mathematical technique of induction and the use of the word induction as per the article to mean deducing knowledge from experience are due to Descartes I believe.[2]

[1] I’m slightly simplifying. Some people teach “the principle of mathematical induction” as a “principle” (whatever that is, but it sounds like an axiom to me), and some people teach it as a theorem like any other theorem but derive it by adding an axiom to set theory called the “well-ordering principle” (that every non-empty set of natural numbers has a smallest element) that turns out to be equivalent to induction. If you don’t accept induction as an axiom and don’t take it to be axiomatic that you can choose the smallest element from a set of natural numbers then as far as I know you can’t do induction in a well-founded way.

[2] https://www.jstor.org/stable/43154967

The problem of induction doesn't need to be solved because the problem refutes itself when time is brought into the picture.

Let's assume Hume's premise "there can be no demonstrative arguments to prove, that those instances, of which we have had no experience, resemble those, of which we have had experience" is true.

Ok, then how do we know this premise will stay true in the future? Because it is true right now? Because every moment in the future will eventually become the present? But how do we know that? Because it has always happened this way. Time has only ever moved forward, never backward, at least in my experience so far. Oops, this is inductive.

If the premise might not stay true in the future, then that means we've managed to acquire a piece of information about the future: the premise of the problem might not be true by then. We've just gained a bit of knowledge about the future which the premise claims we can't possibly do!

I only read up to the "Petals around the Rose" exercise. I found it pretty easy to deduce (induce?) but the author is right, there's no general algorithmic method to do it .. except for intelligence.

So, the solution is "AI", and even then it's only a "probably".