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How to Solve It
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<h2 id="four-principles">Four principles</h2> <p><i>How to Solve It</i> suggests the following steps when solving a <a href="/facts/Mathematical_problem/lFMR51de">mathematical problem</a>: </p> <ol><li>First, you have to <i>understand the problem</i>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li> <li>After understanding, <i>make a plan</i>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li> <li><i>Carry out the plan</i>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li> <li><i>Look back</i> on your work.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> How could it be better?</li></ol> <p>If this technique fails, Pólya advises:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?" </p> <h3>First principle: Understand the problem</h3> <p>"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> depending on the situation, such as: </p> <ul><li>What are you asked to find or show?<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li> <li>Can you restate the problem in your own words?</li> <li>Can you think of a picture or a diagram that might help you understand the problem?</li> <li>Is there enough information to enable you to find a solution?</li> <li>Do you understand all the words used in stating the problem?</li> <li>Do you need to ask a question to get the answer?</li></ul> <p>The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive. </p> <h3>Second principle: Devise a plan</h3> <p>Pólya mentions that there are many reasonable ways to solve problems.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included: </p> <ul><li>Guess and check<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a></li> <li>Make an orderly list<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></li> <li>Eliminate possibilities<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></li> <li>Use symmetry<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a></li> <li>Consider special cases<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a></li> <li>Use direct reasoning</li> <li>Solve an equation<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a></li></ul> <p>Also suggested: </p> <ul><li>Look for a pattern<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a></li> <li>Draw a picture<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a></li> <li>Solve a simpler problem<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a></li> <li>Use a model<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a></li> <li>Work backward<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a></li> <li>Use a formula<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a></li> <li>Be creative<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a></li> <li>Applying these rules to devise a plan takes your own skill and judgement.<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a></li></ul> <p>Pólya lays a big emphasis on the teachers' behavior. A teacher should support students with devising their own plan with a question method that goes from the most general questions to more particular questions, with the goal that the last step to having a plan is made by the student. He maintains that just showing students a plan, no matter how good it is, does not help them. </p> <h3>Third principle: Carry out the plan</h3> <p>This step is usually easier than devising the plan.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work, discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals. <a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> </p> <h3>Fourth principle: Review/extend</h3> <p>Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what did not, and with thinking about other problems where this could be useful.<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a><a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem. </p> <h2 id="heuristics">Heuristics</h2> <p>The book contains a dictionary-style set of <a href="/facts/Heuristics/BcBEkv9c">heuristics</a>, many of which have to do with generating a more accessible problem. For example: </p> <table><tbody><tr><th>Heuristic</th><th>Informal Description</th><th>Formal analogue [<i>original research?</i>]</th></tr><tr><td><a href="/facts/Analogy/LYcGaTdm">Analogy</a></td><td>Can you find a problem analogous to your problem and solve that?</td><td><a href="/facts/Map_(mathematics)/FABF1l2f">Map</a></td></tr><tr><td>Auxiliary Elements</td><td>Can you add some new element to your problem to get closer to a solution?</td><td><a href="/facts/Extension_(predicate_logic)/ruBkmsJ6">Extension</a></td></tr><tr><td><a href="/facts/Generalization/v0b1pwUx">Generalization</a></td><td>Can you find a problem more general than your problem?</td><td><a href="/facts/Generalization/v0b1pwUx">Generalization</a></td></tr><tr><td><a href="/facts/Induction_(philosophy)/EJ992JJ9">Induction</a></td><td>Can you solve your problem by deriving a generalization from some examples?</td><td><a href="/facts/Induction_(philosophy)/EJ992JJ9">Induction</a></td></tr><tr><td>Variation of the Problem</td><td>Can you vary or change your problem to create a new problem (or set of problems) whose solution(s) will help you solve your original problem?</td><td><a href="/facts/Search_algorithm/eJuw0nyz">Search</a></td></tr><tr><td>Auxiliary Problem</td><td>Can you find a subproblem or side problem whose solution will help you solve your problem?</td><td><a href="/facts/Subgoal/N5UtLKVk">Subgoal</a></td></tr><tr><td>Here is a problem related to yours and solved before</td><td>Can you find a problem related to yours that has already been solved and use that to solve your problem?</td><td><a href="/facts/Pattern_recognition/o1s38opj">Pattern recognition</a> <a href="/facts/Pattern_matching/CEwptMs9">Pattern matching</a> <a href="/facts/Reduction_(complexity)/IVWF7a1c">Reduction</a></td></tr><tr><td><a href="/facts/Special_case/bKaJ85XJ">Specialization</a></td><td>Can you find a problem more specialized?</td><td><a href="/facts/Special_case/bKaJ85XJ">Specialization</a></td></tr><tr><td><a href="/facts/Decomposition_(computer_science)/oAQ7l0VA">Decomposing</a> and Recombining</td><td>Can you decompose the problem and "recombine its elements in some new manner"?</td><td><a href="/facts/Divide_and_conquer_algorithm/pC1X7Ws7">Divide and conquer</a></td></tr><tr><td><a href="/facts/Working_backward_from_the_goal/St5dtGYq">Working backward</a></td><td>Can you start with the goal and work backwards to something you already know?</td><td><a href="/facts/Backward_chaining/St5dtGYq">Backward chaining</a></td></tr><tr><td>Draw a Figure</td><td>Can you draw a picture of the problem?</td><td><a href="/facts/Diagrammatic_Reasoning/7twYcR3a">Diagrammatic Reasoning</a><a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a></td></tr></tbody></table> <h2 id="influence">Influence</h2> <ul><li>The book has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.</li> <li><a href="/facts/Marvin_Minsky/HTU5MYLL">Marvin Minsky</a> said in his paper <i>Steps Toward Artificial Intelligence</i> that "everyone should know the work of George Pólya on how to solve problems."<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a></li> <li>Pólya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for <a href="/facts/Mathematics_education/6hXltmmF">mathematics education</a>.<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a></li> <li>Russian inventor <a href="/facts/Genrich_Altshuller/FRbehS7B">Genrich Altshuller</a> developed an elaborate set of methods for problem solving known as <a href="/facts/TRIZ/6Lw9YjMK">TRIZ</a>, which in many aspects reproduces or parallels Pólya's work.</li> <li><i><a href="/facts/How_to_Solve_it_by_Computer/xyI6FCHP">How to Solve it by Computer</a></i> is a computer science book by R. G. Dromey.<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a> It was inspired by Pólya's work.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Inventor%2527s_paradox/qevm3dD4">Inventor's paradox</a></li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/George_P%25C3%25B3lya/cYgGdvJu">Pólya, George</a> (1957). <a href="https://archive.org/details/howtosolveitnewa00pl"><i>How to Solve It</i></a>. Garden City, NY: Doubleday. p. <a href="https://archive.org/details/howtosolveitnewa00pl/page/253">253</a>.</li></ul> <h2 id="external-links">External links</h2> Wikiquote has quotations related to <i>George Pólya</i>. <ul><li><a href="http://www.math.utah.edu/~alfeld/math/polya.html">More information on Pólya can be found here.</a> <a href="https://web.archive.org/web/20120204044137/http://www.math.utah.edu/~alfeld/math/polya.html">Archived</a> 2012-02-04 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li> <li><a href="http://www.softpanorama.org/Bookshelf/Classic/polya_htsi.shtml">Softpanorama page about the value of the book in programming</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Pólya, George (1945). How to Solve It. Princeton University Press. ISBN 0-691-08097-6. {{cite book}}: ISBN / Date incompatibility (help) <a href="0-691-08097-6" target="_blank">0-691-08097-6</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Pólya 1957 pp. 6–8 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Pólya 1957 pp. 8–12 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Pólya 1957 pp. 12–14 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Pólya 1957 pp. 14–15 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Pólya 1957 p. 114 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Pólya 1957 p. 33 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Pólya 1957 p. 214 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Pólya 1957 pp. 8–12 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Pólya 1957 p. 99 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Pólya 1957 p. 2 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Pólya 1957 p. 94 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Pólya 1957 p. 199 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Pólya 1957 p. 190 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Pólya 1957 p. 172 Pólya advises teachers that asking students to immerse themselves in routine operations only, instead of enhancing their imaginative / judicious side is inexcusable. - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Pólya 1957 p. 108 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Pólya 1957 pp. 103–108 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Pólya 1957 p. 114 Pólya notes that 'human superiority consists in going around an obstacle that cannot be overcome directly' - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Pólya 1957 p. 105, pp. 29–32, for example, Pólya discusses the problem of water flowing into a cone as an example of what is required to visualize the problem, using a figure. - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Pólya 1957 p. 105, p. 225 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Pólya 1957 pp. 141–148. Pólya describes the method of analysis - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Pólya 1957 p. 172 (Pólya advises that this requires that the student have the patience to wait until the bright idea appears (subconsciously).) - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Pólya 1957 pp. 148–149. In the dictionary entry 'Pedantry & mastery' Pólya cautions pedants to 'always use your own brains first' - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Pólya 1957 p. 35 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Pólya 1957 pp. 8–12 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>Pólya 1957 p. 36 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>Pólya 1957 pp. 14–19 - Pólya, George (1957). How to Solve It. Garden City, NY: Doubleday. p. 253. <a href="https://archive.org/details/howtosolveitnewa00pl" target="_blank">https://archive.org/details/howtosolveitnewa00pl</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>"Diagrammatic Reasoning site". Archived from the original on 2009-06-19. Retrieved 2006-02-27. <a href="https://web.archive.org/web/20090619082112/http://zeus.cs.hartford.edu/~anderson/" target="_blank">https://web.archive.org/web/20090619082112/http://zeus.cs.hartford.edu/~anderson/</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>Minsky, Marvin. "Steps Toward Artificial Intelligence". Archived from the original on 2008-12-31. Retrieved 2006-05-17.. <a href="/wiki/Marvin_Minsky" target="_blank">/wiki/Marvin_Minsky</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> <li id="fn:30"><p>Schoenfeld, Alan H. (1992). D. Grouws (ed.). "Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics" (PDF). Handbook for Research on Mathematics Teaching and Learning. New York: MacMillan: 334–370. Archived from the original (PDF) on 2013-12-03. Retrieved 2013-11-27.. <a href="/wiki/Alan_H._Schoenfeld" target="_blank">/wiki/Alan_H._Schoenfeld</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li> <li id="fn:31"><p>Dromey, R. G. (1982). How to Solve it by Computer. Prentice-Hall International. ISBN 978-0-13-434001-2. <a href="978-0-13-434001-2" target="_blank">978-0-13-434001-2</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li> </ol>