[A journal entry by Alex Hernandez.]
A conjecture is informally, a guess, a hypothesis and a deduction. Of course you're going to use this when you are problem solving. How I might use one is very simple: Trying to answer a problem. For example, let's say we have the funny furniture problem. Well, at first have an idea of what to do, (just turn it around a bunch of times and then it eventually will be next to itself). However, then you are given information restriction of the chair. That is when you start to really think and understand what you are working with. In this case, you know you can't turn it except in 90 degree angles. You have an idea that it can be done, so you don't give up on the problem yet. Then you start to think about in what ways you can turn the chair: left, right, forwards and backwards. This information becomes very helpful to you, and you start playing around with it. Once you realise certain methods do not work, you begin to not waste your time so much on similar methods; you have deduced that based on certain patterns that other similar patterns may not work. However, if all else fails, you will try the similar patterns. This is like a fanning inward process. You do a lot at the beginning of trying new things, but eventually, the more you do and earn experience, the more you realise that there are just minor details that need to be changed (not necessarily similar patterns). You notice patterns by deduction as well. If you realise that you can get beside the chair in an odd number of turns, then you'll being to pay more attention to the number of turns you're making. Or if it can be done in no less than eight turns, then you analyse why you can't get any close than eight turns.
Conjuring conjectures is a very natural process (at least I think so) when ever you're dealing with problems because people tend to hate to be wrong or lose. So they want to find a way to be right and have other people think they are right and "know" that they are right, and that's what deduction is about.