《如何解题》读书笔记

                                                How to Solve It  (by G. Polya    波利亚)

                                                               —— A New Aspect of Mathematical Method

 

PART II  HOW TO SOLVE IT  如何解题

 

 

1 . Getting acquainted  熟悉问题

     我应该从哪儿开始?从问题的叙述开始。我能做什么?观察揣摩整个问题,尽量使其清晰而鲜明。暂时先抛开细节。这样做,我能得到什么好处?你会明白问题,使自己熟悉问题,并把问题的目标牢记在脑海中。这样全神贯注地对待问题也会调动起你的记忆力,做好准备去重新联想与问题有关的各点。

2. Working for better Understanding 深入理解问题
     我应该从哪儿开始?还是从问题的叙述开始。当你对问题的叙述已如此清楚并已深深地印入脑海,以致你即使暂时不去看它,你也不怕把它完全忘掉时,你就可以开始下面的工作了。我能做什么? 先把问题的主要部分剖析出来。因为前提与结论是“求证题”的主要部分。未知、已知与条件是“求解题”的主要部分。再把问题中的主要部分都弄一遍,并且要逐个地考虑,轮流地考虑,而且在各种组合中来考虑,同时把每个细节与其它细节联系起来,把每个细节与整个问题联系起来。这么做,我能得到什么好处?你会准备好并弄清楚以后可能起作用的细节。
3. Hunting for Helpful Idea  探索有益的念头
    应该从哪儿开始?从考虑问题的主要部分开始。当主要部分能很清楚地排列出来,想得明明白白(这应归功于你前面的工作)并且也记得住时,这时开始做下一步。怎样进行?从各个方面来考虑你的问题,找出与你现有知识有关之处。从各个方面考虑你的问题。分别突出各个部分,考察各个细节,用不同方法反复审查同一细节。把细节用不同方式组合起来,从不同角度考虑它。试着在每一细节中发现某些新意义,尝试在整个问题中得出某些新解释。从你现有知识中找出与问题有关之处。试想过去在类似的情况下有什么帮过你的忙。在你所考察的内容中,设法找出熟悉的东西来,在你所熟悉的东西中,努力找出有用的东西来。能找出什么?一个有用的念头,也许是个决定性的念头,它能使你一限看出解决问题的途径。念头有什么用?它会给你指出整个或部分解题途径,它或多或少地清楚地向你建议该怎么做。念头多多少少还是完整的。如果你有一个念头,你就够幸运的了。碰上一个不完整的念头怎么办?应该加以考虑。如果它看来有好处,就应该多考虑一会儿。如果它看来是可靠的,你应当确定它能引导你走多远,并重新考虑一下形势。由于这个有益的念头,情况已经变化了。你要从各个方面来考虑新形势并找出它与你现有知识之间的系。再次这样做,还能得到什么好处?如果你走运的话,你或许能找到另一个念头。也许下一个念头会引导你去解决问题。也许在下一个念头以后,你还需要几个有益的念头。也许有些念头会把你引入歧途。无论如何,你应当感谢所有的新念头,感谢那些次要的念头,感谢那些模糊的念头,也感谢那些使模糊念头得以纠正的补充性念头。即使你暂时还没有发现什么有价值的新念头,但如果你对问题的概念更完全了,或者更连贯、更和谐或者更平衡了,那你也应当表示感谢。
4. Carrying Out the Plan 实现计划
    应该从哪儿开始?从引导到解决问题的思路开始。当你感到你已抓住主要的联系,并且自信能提供可能需要的次要细节时,就开始。怎幺做?你对问题应抓得很有把握。详细地进行你以前认为可行的全部代数或几何运算。用形式推理或直接观察检查每一步骤的正确性,或者,如果你能够的话, 两种方法都用。如果你的问题很复杂,你可以分成“大”步骤和“小”步骤,每一大步骤又由几个小步骤组成。首先检查大步骤,以后再检验小步骤。这样做,我能有什么好处?这样提出的解,每个步骤无疑都是正确的。
5. Looking Back 回顾
    应该从哪儿开始?从解答开始,它的每一个细节都应该是完整而正确的,怎么做?从各个方面考虑这个解,找出与你已有知识之间的联系。考虑解的细节,并尝试使它们尽可能地简单;研究解答中较冗长的部分,使它们更短些;试着一眼就看出整个解。试着去改进解的各部分,尝试去改进整个解,使它直观,使它尽量自然地适合于你已有的知识。总结你解题的方法,尝试看出它的要点,并且尝试把它用于其他问题。总结所得结果并试着把它用于其他的问题。这样做,我能有什么好处?你可能找出一个新的更好的解,你可能发现的有趣的事实。无论如何,如果你用这方式养成研究与总结你的解的习惯,将获得某些井然有序的,便于应用的知识,并且你将会提高你解题的能力。

 

怎样解题表 G. Pólya

1. UNDERSTANDING THE PROBLEM
    First. You have to understand the problem.
    What is the unknown? What are the data? What is the condition?
    Is it possible to satisfysatisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundantOr contradictory.
    Draw a figure. Introduce suitable notation.
    Separate the various parts of the condition. Can you write them down?
2. DEVISING A PLAN
    Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.
    Have you seen it before? Or have you seen the same problem in a slightly different form?
    Do you know a related problem? Do you know a theorem that could be useful?
    Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.
    Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?
    Could you restate the problem? Could you restate it still differently? Go back to definitions.
    If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous. problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined,
    how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?
    Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?
3. CARRYING OUT THE PLAN
    Third. Carry out your plan.
    Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?
4. LOOKING BACK
    Fourth. Examine the solution obtained.

    Can you check the result? Can you check the argument?
    Can you derive the solution differently? Can you see it at a glance?
    Can you use the result, or the method, for some other problem?
5. ADDENDUM
    Finally. Think about how one might generalize the problem.