I have recently had a revelation and I have very few people to share it with. It would mean a lot to me if the experienced problem solvers in this forum can pitch in and add to/correct the following paragraph. I am thinking that in case there is more to it, then I would love to hear about it and learn. Kindly do provide your input in case you think there is more to it than what is given below. What makes a contest problem tough is the so called "crux move" as Paul Zeitz calls it or the "insight" that Prof. Schoenfeld refers to in his book 'Mathematical Problem Solving'. A problem to find or to prove requires certain logical statements which when connected will form the solution. Exercise problems seldom have a crux move. All the logical statements are straightforward and that's what makes them easy. On the other hand, a tough contest problem will involve one or two logical statements which are hard to observe. So the idea is to 'play around' with the given data and try to fish out a conjecture which can potentially make the solution straightforward. This approach is seldom taught in high school or heck, even in engineering. This approach seems to be unique only to higher level problem solving in math or higher level education in pure math. Since I am an engineer, I never came across this approach in the context of solving math problems before I looked into problem solving in contests. I have taken this approach in engineering during the time when I was doing research in circuit design but it never occurred to me that the approach is same for solving tough math contest problems! In 1976, thee was a problem in the IMO which asked something to the effect of "break 1976 into smaller positive integers so that their product is maximum". As there are a lot of ways to do it (1976=1+1.....+1=1000+976=.....), it might be initially daunting to figure out a way. Simply finding the maximum product for lower numbers (like the maximum for 9 instead for 1976) will reveal a possibility that all the numbers are either 2, 3, or 4 and that the number of 2s cannot be more than 2 (for example in the case of 10, breaking it to 3+3+4 gives the maximum product). Since 1976=3.658+2, the maximum then turns out to be 2*(3^658) if the conjecture is true. Now you have to try to prove the conjecture and if it turns out true, you solved the problem. A problem's solution, when presented, looks deductive. But the solution in the making is totally inductive. You have to play around with the given data to find out a possible pattern so that conjectures can be made. Those conjectures will then be used to solve the problem. If not, you move on to a different strategy to figure out more patterns to solve the problem. The tricky part of the whole process is to first understand the problem correctly and then, to figure out that crux move/insight/critical conjecture which can possibly help solve the problem. I would truly appreciate it if the members of this forum can pitch in and add to it or correct the above. It is a revelation to me but I am quite sure that it is a very routine thing for a very experienced problem solvers/mathematicians of this forum.