• I need to find a solution to this problem
• I need to find a solution for this problem
• I need to find a remedy for this problem

All three sentences are correct, although the second is less common. Usually, one will find that to is the preposition of choice for the word solution. However, for is preferred in certain cases, and I give two examples to illustrate this:

• A bunch of students are working on a physics problem set in the common room. C and J are done with questions 1 through 4 and are currently working on 5. A just finished solving question 4 and wants to check her work with C and J: "Hey guys, could you show me your solution for number four? I'm not sure I did it correctly."

 
• Joe had just bought a fridge for a dollar from an old lady down the road. Still unable to believe his luck, he wasted no time in setting it up in the corner of the hovel he shared with his brother, Alex. Only after plugging in the fridge did Joe realize that the door wouldn't close properly. He repeatedly slammed the door shut only to have it slowly creak open each time to his chagrin. "Hey, bro," Alex finally said, "calm down. Calm down!" He pulled out a crowbar. "There's a quick solution for this."

• I need to find a solution to this problem
• I need to find a solution for this problem
• I need to find a remedy for this problem

All three sentences are correct, although the second is less common. Usually, one will find that to is the preposition of choice for the word solution. However, for is preferred in certain cases, and I give two examples to illustrate this:

• A bunch of students are working on a physics problem set in the common room. C and J are done with questions 1 through 4 and are currently working on 5. A just finished solving question 4 and wants to check her work with C and J: "Hey guys, could you show me your solution for number four? I'm not sure I did it correctly."

• Joe had just bought a fridge for a dollar from an old lady down the road. Still unable to believe his luck, he wasted no time in setting it up in the corner of the hovel he shared with his brother, Alex. Only after plugging in the fridge did Joe realize that the door wouldn't close properly. He repeatedly slammed the door shut only to have it slowly creak open each time to his chagrin. "Hey, bro," Alex finally said, "calm down. Calm down!" He pulled out a crowbar. "There's a quick solution for this."

• I need to find a solution to this problem
• I need to find a solution for this problem
• I need to find a remedy for this problem

All three sentences are correct, although the second is less common. Usually, one will find that to is the preposition of choice for the word solution. However, for is preferred in certain cases, and I give two examples to illustrate this:

• A bunch of students are working on a physics problem set in the common room. C and J are done with questions 1 through 4 and are currently working on 5. A just finished solving question 4 and wants to check her work with C and J: "Hey guys, could you show me your solution for number four? I'm not sure if I did it correctly."

• Joe just bought a fridge for a dollar from an old lady down the road. Still unable to believe his luck, he wastes no time in setting in up in the corner of the hovel he shared with his brother, Alex. Only after plugging in the fridge did Joe realize that wouldn't close properly. He repeatedly slammed the door shut only to have it slowly creak open each time to his chagrin. "Hey, bro," Alex finally said, "calm down. Calm down!" He pulls out a crowbar. "There's quick solution for this."

• I need to find a solution to this problem
• I need to find a solution for this problem
• I need to find a remedy for this problem

All three sentences are correct, although the second is less common. Usually, one will find that to is the preposition of choice for the word solution. However, for is preferred in certain cases, and I give two examples to illustrate this:

• A bunch of students are working on a physics problem set in the common room. C and J are done with questions 1 through 4 and are currently working on 5. A just finished solving question 4 and wants to check her work with C and J: "Hey guys, could you show me your solution for number four? I'm not sure I did it correctly."

• Joe had just bought a fridge for a dollar from an old lady down the road. Still unable to believe his luck, he wasted no time in setting it up in the corner of the hovel he shared with his brother, Alex. Only after plugging in the fridge did Joe realize that the door wouldn't close properly. He repeatedly slammed the door shut only to have it slowly creak open each time to his chagrin. "Hey, bro," Alex finally said, "calm down. Calm down!" He pulled out a crowbar. "There's a quick solution for this."