There is a great quote of Yitz Herstein:
The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver."
A number of such problems can be found in Herstein's classic book "Topics in Algebra".
Given appropriate background, certain mathematical problems lead us to discover things we otherwise might not, often with elegance and surprise. These problems make us feel more clever than we actually are by our discovery of important and deep ideas. They have the delight and surprise of a competition problem, but reveal an important idea for a theory...and the discovery of the idea is accompanied by a lovely sense of ownership since the "would-be solver" has found it. (I'm thinking about a particular problem from Herstein's book that basically forces the idea of a quotient group on the investigator. I don't reveal which problem this is since I don't want to ruin it.)
Anyhow, the purpose of this question is to collect examples of such problems.
Question: What are some good problems that require the discovery of central and difficult concepts of a theory?
Since such problems usually require background, please include the requisite background with your answer.
Please note: I'm not so sure it is a good idea to post excellent problems along with the concepts they lead to...if a problem does what I want it to, one should be able to discover the idea that is needed by working on the problem. A good answer to this question might omit explicit mention of what idea the problem leads to.