Methods of attacking a problem (1 Viewer)

[frog1944]

Hi,

Every now and then I run into a problem and solve it using the "usual" methods taught, however I then see a counter example where they attack it in a different way, and in some cases solve the problem much faster. E.g. eddie woo on this binomial identity problem (https://youtu.be/-KgkfSAbSn0?list=PL5KkMZvBpo5Bcz-V51UHtlg_eBW-PtQ7_).

How do you find these neat and interesting ways to solve problems?
Are there any methods that are much neater and "smarter" ways of solving various problems than currently taught?
If so, what are the methods?

Thanks

 

[Mahan1]

Hi,

Every now and then I run into a problem and solve it using the "usual" methods taught, however I then see a counter example where they attack it in a different way, and in some cases solve the problem much faster. E.g. eddie woo on this binomial identity problem (https://youtu.be/-KgkfSAbSn0?list=PL5KkMZvBpo5Bcz-V51UHtlg_eBW-PtQ7_).

How do you find these neat and interesting ways to solve problems?
Are there any methods that are much neater and "smarter" ways of solving various problems than currently taught?
If so, what are the methods?

Thanks

Hi frog1944,
well, there is no set general rule that we can assign to every situation. For the most part, it comes from experience. Solving many difficult questions, even beyond extension2, will give you enough experience to tackle most questions.In the nut shell, choose methods that order your question or simplify computation in the question, less working out required.I personally find, engaging with challenging mathematical puzzles like blue eyes problem,https://xkcd.com/blue_eyes.html will help you to think faster and being more creative than others. If you are looking for a book that summarises techniques that you may find useful, The art of problem solving by Alfred S.posamentier and Wolfgang Schulz is a good resource. But I still recommend to access hard and challenging questions and solve problems that require creativity like puzzles or books that are written to prepare students for IMO(choose questions that you can solve with HSC knowledge).

 

[He-Mann]

There are many modes of thoughts that are engaged while solving a problem.​

The "usual" way requires less thought as you know how to go about it and have applied it many times ways i.e. doing a problem in a certain way. There's a moment where you are not satisfied of a solution for several reasons including tedium, too complex, etc. At this point, curiosity could be piqued, you attempt to view this problem from a different perspective (e.g. geometrically instead of algebraic) and this will start engaging your creativity and problem-solving muscles as these thoughts are essentially "new" to you. Introspection is key here as you identify optimisation opportunities to enhance your thinking.

Most of us are lazy. We prefer not too think too much. This is one of the causes of weak problem-solving skills of some students studying MX1 or MX2. e.g. your goal is to do the question and solution length/elegance is not the matter.

A differentiator of great and good problem solvers is that they do not impulsively go with their first solution, they take out more time to think of other pathways then weigh-out each one. They do this often and become better decision makers and more careful thinkers.

Of course, practising doing problems and try to give at least two methods will enhance your problem-solving skills. I would to add an additional point to Mahan's answer talking about puzzles. Certain puzzles are great for isolating a subset of thoughts (say, under logical thinking for the blue eyes puzzle) and this allows you to introspect and enhance this subset as the puzzle forces you to concentrate on it.

Hope this helps!