Back in the day, I used to know every one of the 'problems' that you get in problem solving books. The reason is that I used to collect them and use them to teach numeracy. I realised after a while that the problems were always of a certain type and easily solved. In fact I used to love finding a new type because it would spark off new ideas etc.
My students would of course, try to find new problems and give them to me with the hope of stumping me. The thing is, I would always get them. Now this is not because I am smart (believe me - that ain't it), it is because I have worked out the types and subsequently reduced the types of solutions possible. Once you have twenty under your belt, you really need something a bit tougher.
Why people fail to solve problemsAnyway - many of the problems that you see in workshops or in books are simply based on your ability to make 'unwarranted inferences'. Unwarranted inferences are inferences you make about the rules of a problem. No one states them, your brain just puts them in. The designer of the problem has placed information into the problem structure or description to evoke certain inferences. Your job is to overcome this.
Below are some well known examples. Try and solve them and figure out what the unwarranted inferences are that the problem is trying to get you to make.
1. The sixteen dot problemWithout lifting your pencil, join all 16 dots with six straight lines.
2. Chains and linksA woman has four pieces of chain. Each piece is made up of three links. She wants to join the pieces into a single closed ring of chain. To open a link costs 2 cents and to close a link costs 3 cents. She has only 15 cents. How does she do it?
(I love this one... very satisfying)
SummaryIf you want to master these things then think about 'how' the problem is making you think. Then re-think the boundaries you have placed on your interpretation.
Remember - avoid making unwarranted inferences, draw pictures, keep cool.