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Most mathematicians are familiar with the mathematical problem-solving book How to Solve it, by George Pólya. And for those who are not, especially young mathematicians, I would recommend dropping everything and reading it immediately.

In the foreward of many editions of this book, John Conway attributes to Pólya the advice,

"If you can't solve a problem, then there is an easier problem you can't solve: find it."

Meanwhile, one can also find many people quoting Pólya as having said,

"If you can't solve a problem, then there is an easier problem you can solve: find it."

I have added emphasis for the difference.

The former quotation appears in Conway's foreward to How to Solve it. The latter quotation, meanwhile, appears on the How to Solve it Wikipedia page, and also in many other places. Online, it seems that the quotations are split about 50/50 between these two versions.

My question is, what did Pólya actually say? Are the latter quotes simply mis-quoting Conway's version? Or did Conway make an innovation?

I haven't found Conway's version in Pólya's writing explicitly, although he does have remarks with similar substance in How to Solve it. But there there were evidently many editions of this book, and perhaps I have simply missed the right place.

Personally, I find the Conway version of Pólya's quote more erudite and valuable. One can, of course, always find an easier problem of a problem that one can solve, simply by trivializing it, but that doesn't seem important. What is important, however, is to find an easier-but-still-difficult version of your problem, coming into the boundary between the solved and the unsolved from above.

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    $\begingroup$ Isn't there a History of Science and Mathematics SE? $\endgroup$ – Clarinetist Jan 6 '17 at 15:56
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    $\begingroup$ Well, my question would also fit there, if there is, but I think it is also on-topic here and this is the forum of which I am a member. $\endgroup$ – JDH Jan 6 '17 at 15:59
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    $\begingroup$ @Clarinetist: Yes, here, though only in beta at present. I see no harm in asking it here, however: the audience is probably a good bit larger. (As long as I’m commenting, though, I will say that I don’t at all share the OP’s opinion of How to Solve It; it’s been 50 years or so since I read it, but I distinctly remember finding it basically a compendium of the obvious — harmless, but largely a waste of time.) $\endgroup$ – Brian M. Scott Jan 6 '17 at 16:02
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    $\begingroup$ Perhaps we'll have to ask John Conway. $\endgroup$ – JDH Jan 6 '17 at 16:44
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    $\begingroup$ Completely tangentially to the actual question, but I agree completely the first version is way better as a quote. $\endgroup$ – quid Jan 6 '17 at 17:03
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I will quote exactly what as I see on my version of the book (Second Edition, 1957) on page 114.

If you cannot solve the proposed problem do not let this failure afflict you too much but try to find consolation with some easier success, try to solve first some related problem; then you may find courage to attack your original problem again. Do not forget that human superiority consists in going around an obstacle that cannot be overcome directly, in devising some suitable auxiliary problem when the original one appears insoluble.

From all the sources I've read, it doesn't seem like he explicitly wrote the commonly known quote on his book.

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    $\begingroup$ Yes, as I said, I also wasn't able to find the quote explicitly in Pólya's writing. But meanwhile, Conway uses quotation marks in his foreward; he is definitely attributing those words to Pólya. $\endgroup$ – JDH Jan 6 '17 at 16:06
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Possible (indirect) reference, from source:

The phrases that Pólya uses (and lists in his "short dictionary of heuristic" in How to solve it) have eneterd the language of mathematics and mathematics education: "Could you restate the problem ?": [...] Another suggestion is: if one cannot solve the proposed problem can one find a problem (preferably related to the original!) that one can solve [emphasis mine] ?


In the "How to solve it" list, page xvii, we have :

If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem ?

So, it seems that Polya's original direction was : "search for an easier problem"...

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I once encountered George Polya, going up the elevator in the Stanford Library! I told him what a great movie actor he was. (We had seen several of his instructional videos in my combinatorics class.) At least I got a smile out of him. I hope I am not giving away my age with this!

My edition of HTSI does not have Conway's forward; it is really old, the paper has that funny smell, and inside it says "second edition" and "second printing 1973" from Princeton University Press.

Now for the question: I was unable to find any statement from Polya to "search for an easier problem". However, I did find a section on "If you cannot solve the proposed problem", p. 114. There Polya advises to "try to solve first some related problem". He suggests inventing such a problem.

Besides this, he suggests working a related problem (p. 98), Generalization (p. 108), A problem related to yours and solved before (p. 119), Variation of the Problem (p. 209).

However, he also suggests the opposite, Inventor's Paradox (p. 121), wherein a grander problem that includes the original task is attacked. Here the goal is to expand one's vision and think in new directions.

I hope this helps; Polya may be considered the Sun Tzu of mathematics for this work.

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    $\begingroup$ Your username suggests that perhaps you are giving away your age anyway :) $\endgroup$ – David Jan 10 '17 at 23:40

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