The day after President Kennedy challenged America to land a man on the moon,

… the National Space Agency didn’t suit up an astronaut. Instead their first goal was to

hitthe moon — literally. And just over three years later, NASA successfully smashed Ranger 7 into the moon … It took fifteen ever-evolving iterations before the July 16, 1969, gentle moon landing …Great scientists, creative thinkers, and problem solvers do

notsolve hard problems head-on. When they are faced with a daunting question, they immediately and prudently admit defeat. They realize there is no sense in wasting energy vainly grappling with complexity when, instead, they can productively grapple with smaller cases that will teach them how to deal with the complexity to come.

From The 5 Elements of Effective Thinking.

Some may wonder whether this contradicts my earlier post about how quickly people give up thinking about problems. Doesn’t the quote above say we should “prudently admit defeat”? There’s no contradiction. The quote advocates retreat, not surrender. One way to be able to think about a hard problem for a long time is to find simpler versions of the problem that you can solve. Or first, to find simpler problems that you *cannot* solve. As George Polya said

If you can’t solve a problem, then there is an easier problem that you can’t solve; find it.

Bracket the original problem between the simplest version of the problem you cannot solve and the fullest version of the problem you can solve. Then try to move your brackets.

If you can’t solve a problem, then there is an easier problem that you *can* solve; find it.

I agree, but according to the book quoted in the post, Polya said “there is an easier problem that you can’t solve.”

I did a quick search, and I’ve seen Polya quoted both ways. My hunch is that Polya said “can’t” and was misquoted by people who didn’t fully appreciate what he was saying. Finding a simpler problem that you

cansolve is easy; you can trivialize any problem. Finding a problem that is easier than the original, but still too hard to solve, takes more finesse. It brings you closer to the boundary between what you can and cannot do.I may be wrong about what Polya actually said. If he did say “can,” I think it’s worth also considering his advice substituting “can’t”.

How do you know the problem is easier if you cannot solve it either?

Or what makes it easier then?

Reminds of Fermat’s conjecture…

“The secret of getting ahead is getting started. The secret of getting started is breaking your complex overwhelming tasks into small manageable tasks, and then starting on the first one.”

-Mark Twain

Recursion, again…

The beginning of wisdom is this: Get wisdom. (Proverbs 4:7)

I strongly suspect, that you’re trying to find a simpler problem that COULD be solved, but that you don’t know how to solve yet because you haven’t looked at it. That’s because you’re so focussed on the BIG problem that you don’t know how to solve. It’s another way of saying “Divide And Conquer”.

Godlike genius.. Godlike nothing! Sticking to it is the genius! I’ve failed my way to success.

–Thomas Edison

Let me tell you the secret that has led me to my goal. My strength lives solely in my tenacity.

— Louis Pasteur

What I had that others didn’t was a capacity for sticking to it.

— Doris Lessing

Men give me credit for genius; but all the genius I have lies in this: When I have a subject on hand I study it profoundly.

— Alexander Hamilton

AllanL5: Yes, ultimately you want to find a problem you

cansolve. But by stating it in the negative, you take some pressure off yourself. “I’m not (yet) committing myself to solving this. At this point I’m just formulating other problems I can’t solve.”If I’m sufficiently frustrated I might think “State a problem I can’t solve? That’s easy. I can give you a long list of problems I can’t solve!” It’s a way to trick your complaining brain into doing something productive. :)

Your quote appears in John Conway’s Foreword to a 2004 printing of Polya’s book “How to Solve It”. Conway (page xxi) writes that

I don’t know whether or not this is a witticism by Conway.

I can’t find your quoted text in anything written by Polya, either as “can’t solve” or as “can solve”.

Polya does however give a number of heuristics in his book one of which is: “If you cannot solve the proposed problem, try to solve first some related problem.” This heuristic is repeated in a number of places in his book.

Polya also includes a Dictionary of Heuristics one of which (p. 114) is:

Anyway, I think your point is valid. As you write: “Bracket the original problem between the simplest version of the problem you cannot solve and the fullest version of the problem you can solve. Then try to move your brackets.”

what about “try again later”?

And even Project Ranger, JPL had to rework that several times until they finally got it right. Everything was in such a state of flux, and with Ranger being THE Moon mission at the time, they were constantly inundated with instrument add-on requests, etc.

Ultimately, to regain control of the project, JPL had to say “No more!!” just so they could get a stable set of requirements and the real iteration could begin.

I think Polya just chose a word that wasn’t so good. Should instead be: if you can’t solve a problem, then there is an easier problem that HASN’T been solved…

Re: Duncan’s comment, number 9

I was also puzzled by John D. Cook’s blog post here, quoting Polya as saying that. I’ve been studying his famous book “How to Solve it” (2nd ed), and the meaning of the quote is out of character with the rest of the book (though the phrasing is in character).

What I wanted to add was that Amazon “look inside” provides a full text search, which gives the same result that you note: that quote appears in the forward (by Conway), but not in the text itself. I tried several different fragments, but nothing even remotely similar came up.

But other online sources do quote him as saying it (mostly, with “easier problem that you *can* solve), so it does look like he did say something like that, just not in that book (or not that edition). http://en.wikipedia.org/wiki/How_to_Solve_It http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Polya.html

update: it’s also on google books, which has better search (same findings though): http://books.google.com.au/books?id=X3xsgXjTGgoC

The concept of bracketing is right, but the context is Polya’s “inventor’s paradox”. Ie if there is a problem you can’t solve, is there a more ambitious problem you can solve. This seems paradoxical, as it implies that to solve the new problem you will need to solve the original problem. But, the point is that the more ambitious problem may seem inaccessible before conceiving of it as reachable in this way.

Another thing, yes that is Conway being artful!

Chris, When I was taking my oral exams for my PhD, my advisor asked a simple question that stumped me. Then he said “Let me ask you a harder question.” I started to look like I had a clue and he said “Let me ask you an even harder question.” Then I could answer it.

He really wasn’t asking “harder” questions but more general questions. By putting his original question in a more general setting, he put it where I could recognize it.