Dear @Alexander, both are very important. As @Hanno have stated, you can not solve the problem if previous step of problem formulation was not done. Long way from problem formulation to problem solution is covered by hard work (solving)!

Dear Alexander,

a typical Alex-question. My clear answer is,

- first you have to discover the problem,

- second you have to formulate it,

- third solve it please, thats what´s needed at the end.

Dear @Alexander, both are very important. As @Hanno have stated, you can not solve the problem if previous step of problem formulation was not done. Long way from problem formulation to problem solution is covered by hard work (solving)!

Both are important but first is to diagnose problem so we can lead to solution. Because if we not have a defined problem then we cannot lead to solution.

Dear Sir,identification of the problem is also important after that only one can formulate it for solving. Regards

Sure, dear Hanno and Ljubomir. That is impossible to solve what is not formulated. But, with time passing, the problem formulation is widely recognized and becomes common knowlege, not every one even knows who exactly formulated the problem. Then, importance is shifted toward problem solution. Grigoriy Perelman solved the Puancare problem, his solution was recognized as having utmost importance and advancing the field of Mathematics. 

I agree that both are important; to clearly state a problem contributes to it being solved.  Both are not easy.  And it's not unusual that the solution comes many years later.  It's the way science works, that teaches us the value of patience or perseverance.

Here's another interesting problem.   Why can’t the world’s greatest minds solve the mystery of consciousness?  Philosophers and scientists have been at war for decades over the question of what makes human beings more than complex robots.

http://www.theguardian.com/science/2015/jan/21/-sp-why-cant-worlds-greatest-minds-solve-mystery-consciousness

Both are necessary, but in general, the lack of problem solving ability that plagues the world, in my experience, is due primarily to an inability to formulate the problem to start.

Michkelson formulated the experimental problem to demonstrate that speed of light depends on the speed of source. He failed to prove his statament, as his experiments demonstrated independence of speed of light from the speed of source. Michkelson, personally, refused to recognize his own finding as the law of nature. But his result was correct and exceeded in importance the problem formulation. 

"Say, Capillary Pressure Function in Porous Media problem was formulated by Bukhingham in 1907, but solved in 2014, after 107 years. What is more important ?" - 

i would consider this sort of situation as a very special and very small subset of the overall question area.  - Most situations involve a much shorter period (say, weeks or months).  In most ordinary cases encountered, it is the same person or groups of people formulating and solving a problem, and just getting organized and trying to decide what is needed is the decisive step. 

The special circumstance of a long standing problem is a different-than-usual situation. 

I agree with Aldo, a wise answer. Hanno is correct, my field of specialization on RG is to post questions, I don't have the answer on my own. But RG peers are smart people, always resolve the question in the following discussions. So, I trust in Collective Mind. 

@Sayed. Not always. The Law of Electromagnetic Induction was formulated and partially solved by Faraday. But not untill appearance of the Maxwell Equations the effect was described in full. Maxwell, claiming Electromagnetic Waves, extended Faraday's discovery. So, both, the problem formulation and general solution are great works in Physics, and are of equal significance. 

As the question asks about which of them is more important, and to formulate something is to question one's reality. Solving is important, but never possible if the questioning is not raised.

Dear Vilemar,

thats the reason for my listing.

Sorry Hanno,

Sometimes I go backwards. Answer and then read others.  Fortuantely, your listing came first and it is better "formulated".

Dear Vilemar,

you misunderstood. It was a thanks for the confirmation of my opinion.

Dear All,

Very interesting answers given by RG members. To answer Prof. Alexander's question seems like "Problem" and to answer it as "Solving the Problem".

Thank you Prof. Alexander to formulate this problem.

I think a right formulation is the first step in the problem solution.

@Hanno and Alexander: very interesting question and insightful answers, allow me to put a rewording and offer expanded version of Hanno's answer. The solution may be not as plain and obvious, so we need to check wtih experiment. So my outline is something like this:

A. Problem discovery

B. Problm formulation

C. Finding solution

D. Check with experiment

E. If the experimental result is different from theory's prediction, then 

F. Try to reformulate the problem

G. Test and check again with experiment

Great Hanno,

I was very proud that my answer was similar to yours. But to save people's time I will read the other answers first next time. I was waking up and I thought the question was very intersting...could not go through the other answers...

Dear Victor,

thanks, this improvement of quick ideas is one of the reasons for RG.

And please don´t forget, could be that your primary idea was wrong and you have to find a new model.

Definitely formulation of a problem is an astute observation of theoretical or applied fields of knowledge which is the first act, not to give priority of operations or importance either, for unless a problem is stated there is no what we call a solution. But solving valid problem, unless done by the one who creates it, takes countless of years and unprecedented direction which requires wider domain of knowledge. That is what happens in the case of solving existing mathematical conjectures and hypothesis where new and never existed theories developed. 

That is a good post by Dejenie Lakew. 

This question deserves a quote by Cantor.

"In mathematics the art of proposing a question must be held of higher value than solving it."

(A thesis defended at Cantor's doctoral examination)

Georg Cantor (1845 - 1918)

I didn´t know Juans citation, and to be honest I´m mising the sense of Cantors statement. Could somebody help me please?

PS: I know and feel that asking is the beginning of philosphy and  science. My problem is the classification.

I think that Science is going forward by formulating problems, since by setting the question a whole system of people, institutions and other stuff is starting to work for a reason. The solution will come after a century, but we know a new sector to search about...

In a sense, Einstein special relativity paper, 1905, solved the old problem of elusive Ether, rejecting its existance. 

And I like Brenda's example, where solution preceed the problem.

Dear Brenda, really interesting point of view. But how can you discover a drug (for what?) and later on assign illnesses? Please give some example.

There are problems which are easy to formulate. Particularly problems related with natural generalizations of known results. Also there are problems about relations between concepts that nobody has yet thought about. In mathematics, as a general rule, solutions  are more important. If the solution of a problem is “easy” then almost nobody will appreciate it. Even some people can make a joke on it.

Problem formulation is the creative and probably the more important step towards overcoming a problematic state than problem-solving. A good definition of problem is believed to be more than half of the way towards its eventual elimination. “The formulation of the problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.” ― Albert Einstein

Often problem solution extends initial problem formulation, like Maxwell Equations extended initial Faraday Electromagnetic Induction and Grigoriy Perelman started new field of Mathematics, working on Puancare Junction. 

Dear Brenda,

interesting examples. I now remember a similar case, ASS. Invented as modification of salicyl acid which was used against headaches for hundred of years. Today it has a lot of applications eg against thrombosis.

Problem formulation is the beginning of every problem solving strategy/methodology. Problem formulation is the process by which the Universe of Discourse is analyzed to create a formulated problem, which is sufficiently well defined to enable specific action.The accuracy of problem formulation greatly affects the acceptability and credibility of simulation results.

Sometime the problem solution rejects the problem formulation, as Michkelson Experiment and Einstein Special Relativity rejected idea of Ether. But laws of Dialectics are strong, so, the Ether is back, in the form of Dark Energy. 

Problem formulation is more important than solving. But, both are essential. Leaving any problem without solution is unfair.

We can perform the metaphor of Science~Human Body: In health the first step is always the diagnosis of the illness. Remedy probably would be impossible (example: some kinds of cancer), but a correct diagnosis can make the life of a human more easy...

The same holds in Science: For example we have been searching to detect gravity waves 100 years since 1915 and we cannot find them. Are we sure that our theory for gravity is correct? And what if we actually have not a valid theory and we have just some minor approximations of the truth? If we do not make a diagnosis, then we will spend our money for chimeras...

“If I had an hour to solve a problem I'd spend 55 minutes thinking about the problem and 5 minutes thinking about solutions.” ― Albert Einstein

To get solution to some questions the problematic should be well defined and clearly formulated. A problem good understood holds in itself  the 50% of its resolution, the 50% rest is based on the investigation of the proper solution or optimized approach. However as the scientific knowledge is fundamentally built on the basis of  continuous connections or junctions between fields or disciplines, may be some solutions to a specific question could be a solution to another. At this steps the solution could be or brings a new problematic to a new question that should be identified and clearly formulated.

Problem formulating means you know more. 

In language research, if your research problem is not well stated the solutions that you submit becomes meaningless. In this regard, I think problem formulation is more important that solving the problem.

I highlight your attitude to discuss explicitly in RG your scientific achievements, dear Alexander.

To solve an Engineering problem, you need to have a suitable formulation. The formulation is strongly dependent of the problem, but is also dependent from the technique you use to solve it. I am thinking about an optimization problem, for example, and the mathematical programming technique (or optimization technique) you use to solve it. The technique may require being adapted also to the problem through the formulation, but the problem should be never being adapted to the technique. This basic rule cannot be forgotten. I got excellent results (a little dream) in the three-dimensional optimization of urban drainage systems (sanitary wastewater systems and storm water systems).

So, in my opinion, the problem solution is the focus (a pre-requisite for a researcher is to have capacity to understand the scientific problems). The formulation is the initial or intermediate process or model that you use to solve it.

Formulating a problem needs knowledge obviously, but solving it needs more knowledge.

Both of them are really important  for progress in any sector.

Last news from 'gravity waves detections': sorry error!

http://www.esa.int/Our_Activities/Space_Science/Planck/Planck_gravitational_waves_remain_elusive

@Demetris. Something, which is not measurable, may not exist. This type of dissapointment would not be first, and neither is the last. 

The question is, quit the research or try again to prove theory and models. I´ve no idea.

Let some one with mighty brains, like Grigoriy Perelman, comment on Gravitational Waves. He assured the public in quitting Mathematics, but not Physics. Resolving this problem will make him more famous than rejecting $1 Million monetary prize. What a waist for him not working on a hard problem. 

If we count the amount of money that we have spent till now on our efforts to prove by force that our current theory of gravity is correct, then we could save even millions of children from death in Africa. So, the 'problem formulation' is almost critical, among others, for human lifes.

@Demetris. Let Grigoriy Perelman resolve Gravitational Waves problem. Then let him donate $1 Million Nobel Prize money for children of Africa, instead of stupid rejection.

I wish that Grigoriy Perelman will start thinking about Gravity and I will be happy to solve the puzzle, since I am sure that he will donate the money of whatever prizes to philanthropy.!

Grigoriy Perel'man committed utmost stupidity, rejecting Prize flat, not donating to any case. But he is the best bet, for solving gravity problem. 

President Putin defined Perelman as the Pride of Russia, man of highest moral standads. When some people in Russian Government came with idea to enforce Perelman in leading decent live, above poverty line, Putin vitoed this idea. He told, that we are mere people, can't judge our best Scientist, who is genius. May be, he added, this is his way to concentrate on another great science problem. 

Just a small advice: If he accept the proposal, please tell him to start from the 'apple of Newton' and not from the 'elevator of Einstein'...

Dear Prof. Alexander J. Babchin,

Formulation of problem is a way to reach the sustainable solution.

Sure, Afaq, but some times even wrong formulation, like Ether, can lead to great solutions, like Relativity. 

Kindly yours,

Alexander

Dear George,

formulations and conjectures are no synonymes. How will you solve a problem without defining and formulating it? You won´t see the problem at all. But of course you can use some conjectures in your theory and discard them by trial and error.