What is the difference between "solution of problem" and "solution to problem"?

Question

Could someone explain, please, what is the difference between "solution of problem" and "solution to problem"?

Answer 1

In most dictionaries, such as LDOCE, MW or ODO you will find that typical/recommended preposition is to or (somewhat rarer) for.

Macmillan dictionary gives this usage note:

The usual preposition to use with solution is to, not of:

✗ The role of the government is to find the solution of this problem.

✓ The role of the government is to find the solution to this problem.

You can also use for, but this is much less common:

To find the best solutions for managing waste, all these options have to be considered.

The preposition of is used with solution mainly in technical contexts, when it refers to a liquid with another substance dissolved in it:

The containers are cleaned with a weak solution of caustic soda.

Note that this quote doesn't say explicitly that solution of a problem is ungrammatical, it just states that the usual preposition used here is to.

Online Oxford Collocation Dictionary also lists to and for as prepositions typically used after solution (meaning an answer, not a mixture of liquid with dissolved substances).

But!

This doesn't say that using solution of the problem is absolutely impossible. The meaning of the word of would be:

used to show who something or someone belongs to or has a connection with: (LDOCE)

that is, this is a corresponding solution to/for the given problem.

There is an example of usage in Oxford Advanced Learner's Dictionary:

Will this lead to a peaceful solution of the conflict?

(but it took a lot of digging to find this one).

There are examples in books:

We then apply the backward Euler method and the trapezoidal method to the solution of the problem (4.30).

from: Numerical Solution of Ordinary Differential Equations by Kendall Atkinson, Weimin Han, David E. Stewart, John Wiley & Sons, 2009

(Of was probably used instead of to for avoiding repetition.)

The formal solution of the problem is then executed and the modeler has the task of interpreting the solution in terms of the model.

from: Tissue Mechanics by Stephen C. Cowin, Stephen B. Doty, Springer Science & Business Media, 2007

---

In conclusion: I would definitely advise that you use solution to the problem, since this is the most commonly used option. See this Google Ngram. This construction sounds more idiomatic (to me) than the other two. If you need to avoid repetition of 'to', you can say solution for the problem. Although the construction solution of the problem wouldn't be my first choice, I would refrain from calling it incorrect, or ungrammatical.

Oh, and to answer your actual question about the difference in meaning: there is none.

Answer 2

BNC has about 500 incidents of "solution of". Quite a number are mathematical solutions of equations. But you also find "solution of a problem". So one can only say "solution to a problem" is more frequent, but "solution of a problem" is a possible variant. There is no logic or grammatical reason that makes "solution to a problem" the only possibility. If dictionaries often give only "solution to" it is a mere statistical thing, and it is imprecise.

http://bnc.bl.uk/saraWeb.php?qy=solution+of&mysubmit=Go

And for how long has America exerted pressures to prevent a convening of an international conference for a peaceful solution of the Middle East conflict. - BNC, solution of, ARW 461

Answer 3

"Solution of the problem" sounds wrong here because it would imply that the solution came out of the problem, which is not the case. The solution to the problem is the better sounding version of the phrase.

Let's take another similar phrase "result of the war", here of is the correct usage since the result is coming out of the war.

Another example of using "to" would be "he's the Romeo to my Juliet." Use to when you are pairing two things together, but one doesn't necessarily come out of the other.