The 9 Dots Problem
The best hint, in my view, is to mention that often, when solving a problem, we implicitly impose constraints that have not come with the problem. One classical example is where nine dots are arranged on the sides and the center of a square as in the picture below.
The solution lies in the observation that it's permissible to cross square boundaries.
Now, try to think of a restriction you imposed on yourself which was not inherent to the problem.
As an aside, Lars Hellvig from Stockholm, Sweden picks up this line of thought and points out even more of self-imposed restrictions that I missed considering the 9-point problem above.
Xinding Sun from University of California, Santa Barbara found yet another twist to the problem.
Further variants are discussed at the CTK Exchange - the old archive.