Thinking lazily about problem-solving methods

A little while ago I saw this somewhat unusual list of words:

It appeared on the xkcd blag, and Randall shared very little about it: It was in his handwriting, it looked faintly familiar, and he had no idea what it was about. You can see the whole story here.

I whiled away some time just thinking about how I would go about solving it.

That took me back a while, letting me reminisce about problem-solving techniques in general, how I learnt about them, what I found enjoyable about that learning.

For example, I still remember the first time I was presented with the “n” players in a knockout tournament, no ties, how many matches in the tournament question. n was set at 128; some people were doing the traditional 64+32+16+..+ thing, the rest of us had listened to the teacher. He very pointedly said “Think. How many losers?”. And we were ecstatic children, discovering for ourselves the truth that each match produced precisely one loser, and that the tournament needed 127 people to lose….

And so I thought to myself, of all the problems I’d seen where there was a lesson in problem-solving to be learnt, which one had I enjoyed the most?

I decided my personal winner was this one, presented to me in two parts:

(a) There is one, and only one, ten-term arithmetical progression of primes between 1 and 3000. Find it.

(b) Prove that no arithmetical progression of primes, containing eleven or more terms, can possibly exist between 1 and 20000.

I may have got the precise wording wrong, but the salient numbers are correct. See what you think. And if you have similar examples, where you learnt a simple problem-solving technique that stayed with you for the rest of your life, then please do share it.

[Incidentally, I still have no idea what Randall’s list is about. I used a progressive google search approach on the words (where you add one search term at a time and see how the top results behave) and the best I could come up with was “words at the top of successive pages in Animal Farm“. But it feels lame and unworthy. I quite liked one of the suggested answers, that the words were a sequence of captcha words. When I last looked not even Randall had figured it out.]

6 thoughts on “Thinking lazily about problem-solving methods”

  1. An arithmetical progression of primes!
    I did not expect to see that topic appear in this blog, and it makes me wonder:

    1- what percentage of the readers of this blog are able to solve it? (I must admit I only discovered half the answer i.e. 210 before googling it)

    2- what percentage of the readers of this blog are interested in number theory vs the global blogger population?

    3- in what context did JP hear about this problem? Was this part of his curriculum?
    If so, is number theory widely taught in some regions of the world or is JP a mathematician or is Belgium an exception?

    4- what does this test reveal about those who pass it — or those who ask it? It reminds me of a (much simpler) question that was traditionally asked in my company during the hiring interviews: what is the mass of the earth (give or take a factor 10), which led into sub-questions when the candidate was solving it; the one I remember is “why is the diameter 12000 and some kilometer?” (answer: by definition of the meter, which was 1:10,000,000 of a meridien)

    So this entry probably tells a lot about JP and his readers, but I do not know what…

  2. Andrew, loved the truck!

    Benoit, there are many questions you ask that I have no answer to: number of readers, what they are interested in, etc.

    What I can say is this. I came across the problem while at school, probably in a “mathematical olympiad” paper the teacher had circulated around the class. It did not form part of the curriculum per se, but for sure we were encouraged to study number theory, and primes within that.

    [incidentally, the school in Calcutta was run primarily by Belgian (!) Jesuit priests in those days.]

    I am not a mathematician; while I studied maths at school (and within my economics and statistics courses at university), all I can say is that I’ve had a lifelong interest in primes.

    I have no idea what this reveals about people who ask or answer such questions.

    So why did I share it? I found it elegant, beautiful, fascinating. And I like sharing things that I find elegant, beautiful, fascinating. And people seem to come back and read more…..

  3. That blag post is still annoying me today – I’m not much of a fan of unsolvable mysteries…

    As for the arithmetical progression of primes, I must admit I haven’t quite figured that out yet. A hint maybe, before I resort to invoking Google…
    Kinda reminded me of the 8 ball problem though…

  4. FND: Hint as requested. Part (b) asks for proof that no AP of 11 or more primes as terms can exist between 1 and 20000. 11 or more terms means 10 or more “constant differences”. If you could prove that the minimum constant difference required in such a series, when multiplied by ten, exceeded 20,000, then you would solve part (b).

    So the solution (to both problems) lies in understanding what kind of characteristics the constant difference needs to have, then working out how to determine what it could be.

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