Sunday, April 02, 2006

How to solve it?

          (by G. Polya)

Getting Acquainted
Where should I start? Start from the statement of the problem.

What can I do? Visualize the problem as a whole as clear and as vividly as you can. Do not concern yourself with details for the moment.

What can I gain by doing so? You should understand the problem, familiarize yourself with it, impress its purpose on your mind. The attention bestowed on the problem may also stimulate your memory and prepare for the recollection of relevent points.

Working for Better Understanding
(Questions are: What is the unknown? What are the data? What is the condition?)

Where should I start? Start again from the statement of the problem. Start when this statement is so clear to you and so well impressed on your mind that you may lose sight of it for a while without fear of losing it altogether.

What can I do? Isolete principal parts of your problem. The hypothesis and the conclusion are the principal parts of a "problem to prove"; the unknown, the data, and the conditions are the principal parts of a "problem to find." Go through the principal parts of your problem, consider them one by one, consider them in turn, consider them in various combinations, relating eache detail to other details and each to the whole of the problem.

What can gain by doing so? You should prepare and clarify details which are likely to play role afterwards.

Drow a figure. Introduce suitable notation. Separate the parts of the condition. Can you write them down?

Hunting for the Helpful Idea
(Questions are: Have you seen it before? Do you know a related problem? Here is a problem related to yours and solved before. Could you use it?)

Where should I start? Start from the consideration of the principals parts of your problem. Start when these principal parts are distinctly arranged and clearly conceived, thanks to your previous work, and when your memory seems responsive.

What can I do? Concider your problem from various sides and seek contacts with your formery acqired knowledge.
Consider your problem from various sides. Emphasize different different parts, examine different details, examine the same details repeatedly but in different ways, combine the details differently, approach them from different sides. Try to see some new meaning in each detail, some new interpretation of the whole.
Seek contacts with your formery acquired knowledge. Try to think of what helped you in similar situations in the past. Try to recognize something familiar in what you examine, try to perceive somthing useful in what you recognize.

What could I perceive? A helpful idea, perhaps a decisive idea that shows you at a glance the way to the vary end.

How can the idea could be helpful? It shows you the whole of the way or a part of the way; it suggest to you more or less distinctly how can proceed. Ideas are more or less distinctly how you can proceed. Ideas are more or less complete. You are lucky if you have any idea at all.

What can I do with an incomplete idea? You sould consider it. If it looks advantageous you sould concider it longer. If it looks reliable you should ascertain how far it leads you, and reconsinder the situation. The situation has changed, thanks to your helpful idea. Consider the new situation from various sides and seek contacts with your formery acquired knowledge.

What can I gain by doing so again? You may be lucky and have anoder idea. Perhaps your next idea will lead you to the solution right away. Perhaps you need a few more helpful ideas after the next. Perheps you will be more led astray by some of yours ideas. Neverthe less you should be grateful for all new ideas, also for the lesser ones, also for the hazy ones, also for the supplementary ideas adding some precision to a hazy one, or attempting the correction of a less fortunate one. Even if you do not have any appreciable new ideas for a while you should be grateful if your conception of the problem becomes more complete or more coherent, more homogeneous or better balanced.

Carrying Out the Plan
(Questions are: Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove thet it is correct?)

Where should I start? Start from lucky idea that led you to the solution. Start when you feel sure of your grasp of the main connection and you feel confident that you can supplay the minor details that may be wanting.

What can I do? Make your grasp quite secure. Carry through in detail all the algebriac or geometric operations which you have recognized previously as feasible. Convince yourself of thr correctnrss of each step by formal reasoning, or by intuitive insight, or both ways if you can. If your problem is very complex you may distinguish "great" steps and "small" steps, each great step being composed of several small ones. Check first the great steps, and get down to the smaller ones afterwards.

What can I gain by doing so? A presentation of the solution each step of which is correct beyond doubt.

Where should I start? From the solution, complete and correct in each detail.

What can I do? Consider the solution from various sides and seek contacts with your forery acquired knowlidge.
Consider the details of the solution and try to make them as simple as you can; survey more extensive parts of the solution and try to make them shorter; try to see the whole solution at a glance. Try to modify to their advantage smaller or larger parts of the solution, try to improve the whole solution, to make it intuitive, to fit it into formery acquired knowledge as naturally as possible. Scrutinize the method that led you to the solution, try to see its points, and try to make use of it for other problems. Scrutinize the result and try to make use of it for other problems.

What can I gain by doing so? You may find a new and better solution, you may discover new and interesting facts. In any case, if you get into the habit of sureveying and scrutinizing your solutions in this way, you will acquire some knowledge well orderedand ready to use, and you will develop your ability of solving problems.

Nice illustration could be found here

Next step is, How to work?

These days our lives are busier than ever. We work more than ever. We are more stressed and exhausted than ever before. And yet we get less done and are not as happy.

It doesn’t have to be that way.

The problem is that we are overloaded with information and tasks, and we try to get everything done instead of just the most essential things. Solution: focus on only the essential, eliminate the rest, and allow yourself to get into that beautiful state known as “flow”.

And although it can be hard to give up all the busy-ness that we’ve grown accustomed to, the change will have tremendous benefits on our sanity, our stress levels, our happiness, and yes, our productivity.

Here are 10 simple ways to be more productive with less effort:

1. Clear your head. It’s impossible to gain perspective, and to know what is truly essential, if we are in the middle of an information stream. Take an hour, or half a day if possible, to shut off the information flow, and to get a larger view of your life and your job. The time you take off will be well worth it. Tell everyone that you are unavailable, shut off all communications, shut yourself in somewhere private, and take some time to think about what is important. What do you want? Where are you going? What will it take to get there? Another good way to clear your head, which is necessary for focus, is to write down everything that you need to do, all your tasks and projects and ideas. Dump the contents of your mind on paper, and then stop thinking about them for a little while.

2. Focus on the essential tasks. Once you’ve gotten your head cleared, you need to figure out what tasks are most essential. Ask yourself this magic question: “What task can you do that will get you the most return on your time?” Figure out the project that will get you the most recognition, win you awards, or get you the most business. Something that will pay off big. Not something you’ll forget about in a week, but something that others will remember you by. This is an essential task. Make a list of these types of tasks — they’re your most important things to do this week.

3. Eliminate the rest. Now look at your overall list. What’s on there that’s not essential? Can you just drop them from your schedule? Or delegate them to someone else? If not, put them on a “waiting list”. Then, as you focus on your essential tasks, check back on this waiting list every now and then. Sometimes you’ll realize that the less essential tasks weren’t really necessary at all.

4. Do essential tasks first. If you’ve got a list of things to do today, and one or two of them are truly essential, do those items first thing in the morning. Don’t wait until later in the day, because they’ll get pushed back as other urgent stuff comes up. Get them out of the way, and your productivity will truly soar.

5. Eliminate distractions. You can put essential stuff on your list all year long, but if you are constantly interrupted by email notifications, IM, cell phones, your RSS reader, gadgets and widgets, social media, forums and the like, you’ll never be productive. Turn these things off, disconnect yourself from the Internet if possible, clear your desk of all papers, clear your walls and surrounding areas, and allow yourself to truly focus.

6. Use simple tools. Don’t fidget with a bunch of gadgets or the latest and coolest applications. Find a simple notebook for writing things down, a simple to-do list (no frills) and the simplest application possible for doing your work. Then forget about the tools and think only of the task at hand. If you’re too worried about the tools, you’re not actually doing anything.

7. Do one thing at a time. Multi-tasking is a waste of time. You can’t get things done with a million things going on at once, pulling for your attention. Focus on the essential task in front of you, to the exclusion of all else, and you are much more likely to get it completed, in less time, with less effort.

8. Find quiet. In addition to a quiet working environment, you need time every day that you can call your own, where you don’t have to do work. This could be through reading, taking a bath, walking in nature, going swimming at the beach, going jogging, meditating. Not reading your feeds. Get away from the information overload and find that peace that will allow you to truly focus when you do work, and to review your day in your mind, and to get the perspective to see what is essential.

9. Make the most of your work. It’s one thing to write something great, or to create something fantastic. But it’s entirely another thing to make that great thing explode, to get you attention, to earn the recognition you deserve — which will lead to more business or more opportunities. Once you’ve created the Next Great Thing, promote it, show it to others, find a way to have it carry you as far as it can take you. Don’t just create something and move on to the next thing. Use your energy and talents to their fullest extent.

10. Simplify some more. Once you’ve simplified down to the essential, and eliminated distractions, you should become productive. But distractions and the unnecessary have a way of creeping back in and accumulating. Every now and then, take a look at what you’re doing, at the information coming into your life, at how you spend your time and the tools you use. Then simplify some more.


Anonymous said...

What about wishful thinking that is described in SCIP? When approaching a new kind of problem, allow yourself to conceive of a construct, like a helper procedure, that could handle a reduction of the problem, and then proceed on planning the reduction of the problem as though the construct actually existed. This allows you to defer the challenge of actually implementing such a construct until later, when you’ll have had time to refine your understanding of exactly what your construct has to do. This kind of thinking is itself kind of iterative, and it also mirrors the design strategy of breaking big procedures down into smaller ones.

Anonymous said...

Here is a interesting link that do the parallel between "How to solve it?" and programming (Haskell)

algorithms (1) cpp (3) cv (1) daily (4) emacs (2) freebsd (4) java (3) javascript (1) JSON (1) linux (2) Lisp (7) misc (8) programming (16) Python (4) SICP (1) source control (4) sql (1) думи (8)