gerund, infinitive or something else

[jimbean]

Examples:
I was trying to figure out my statistics homework when I discovered a shortcut to solving the problems.
I was trying to figure out my statistics homework when I discovered a shortcut to solve the problems.

First, are both correct, to solve and to solving? If not, why?
Second, is solving the object of the preposition to? and what is its function?
Third, is to solve an infinitive (acting as an adjective)?

Thanks.

[susiedq]

Better: I was trying to figure out my statistics homework when I discovered a shortcut for solving the problems.


I was trying to figure out my statistics homework when I discovered a shortcut to solve problems.

[philo2009]

Examples:
I was trying to figure out my statistics homework when I discovered a shortcut to solving the problems.
I was trying to figure out my statistics homework when I discovered a shortcut to solve the problems.

First, are both correct, to solve and to solving? If not, why?
Second, is solving the object of the preposition to? and what is its function?
Third, is to solve an infinitive (acting as an adjective)?

Thanks.

The second, employing an adnominal (or adjectival) infinitive phrase, is rather unnatural, and some speakers would not accept it. The first, employing an adnominal prepositional phrase, is both correct and natural (although 'for solving', as suggested by another contributor, would also be possible).

Yes, gerund 'solving' here functions as the object of the preposition 'to', whilst simultaneously governing NP 'the problem' as its direct object.

[hoangkha]

- There are no short cuts to economic recovery. (OALD)
TO in this construction is a preposition.
So
- I was trying to figure out my statistics homework when I discovered a short cut to solving the problems.(correct)
- I was trying to figure out my statistics homework when I discovered a short cut to solve the problems..(incorrect)

[5jj]

- I was trying to figure out my statistics homework when I discovered a short cut to solve the problems. (incorrect)

As philo wrote, that "is rather unnatural, and some speakers would not accept it".