As coordinator of the AlgPoW service, I am happy to inform all teachers who are using my problems with their students, that I have just add some significant new material to my Guidelines for Writing PoW Answers page. The most important item concerns my policy regarding the use of guess-and-check as the primary strategy for solving the problems. It says:
Use of Guess-and-Check Procedure
In general, the method of "guess-and-check" is not allowed in AlgPoW as your primary strategy to solve the problem. This is not saying guess-and-check is not a good way to solve problems. In fact, it is often a good way to start to understand a problem, and therefore recommended for that. But for most of our problems, you must define variables or unknowns, then form equations to solve by logical steps.
Historically, it was the main way that problems were solved. But as advances were made in symbolic notation, mathematicians moved away from it and toward the more efficient and time-saving methods of step-by-step manipulations on equations.
One of my mentors told me that he advises students in the following way:
"Guess and check is a valid problem-solving approach. However, it also one of the most difficult to explain. If you are going to use guess and check, you must list every guess, along with the reason that you know the answer is incorrect. You also must explain why you know your final answer is the only possible answer. In all, a pretty long process; however, since this is the Algebra Problem of the Week, you might want to try algebra. Please read the "Guidelines for Writing POW Answers." The link is at the bottom of the problem."
So, unless otherwise indicated, please do not use guess-and-check as your principal solving procedure. *********
I hope this makes things more clear for everybody. I would like to recommend that copies of the "new and improved" Guidelines page be made for distribution to students. Some teachers have already done this, but now there is more information than originally.