grammar - 'To solve' versus 'To solving'

Trying to understand what seems to be a very subtle difference in written and spoken English. Recently, I've seen articles that use 'to + gerund' and 'to + infinitive' in the exact same situations, especially when preceded by an independent clause with a subjective complement:

Editor's Corner: There is no silver bullet to solve homelessness

'There is no silver bullet' to solving N.D.'s worker shortage, officials say

These exist in relatively equal distribution on google, (16k for 'no silver bullet to solve' and 13k for 'no silver bullet to solving), and I'm having a tough time understanding syntactically what might be going on here.

I thought a good analogy might be the "I look forward to hear/hearing from you" mistake than many English language learners make, but this doesn't seem to be that same situation, both in the sense that to my native ears one doesn't sound much better than the other, and I'm not sure they're the same syntactically.

In my (probably misguided) syntactic reading of the first sentence ('There is no silver bullet to solve homelessness'), I think that 'to solve homelessness' is an infinitive clause acting as a prepositional phrase:

And therefore is syntactically similar to something like, "There's no wrong way to eat a Reese's." This, to me, seems correct. Does this seem on base? And if so, why do these two different approaches exist equally on the net? Is there any potential syntactic reading for 'to solving' that makes sense here?

Answer

The infinitive (or perhaps it should be analyzed as the subjunctive?) gives a sense of completion, while the gerund gives more of a sense of process. So "There is a new approach to solving homelessness" indicates that the approach will help work on the problem of homelessness, while "There is a new approach to solve homelessness" implies that it will successfully deal with homelessness. A further complication is that "silver bullet" is a metaphor (or, at least, I hope people aren't discussing literally solving homelessness with silver bullets), and there can be differences as to what people consider it a metaphor for. It's often used as a metaphor for something that completely takes care of something, which suggests the infinitive. If, however, it's used as a metaphor for a tool used in the process of doing something, rather than something that automatically gets one to one's end goal, that suggests the gerund.