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edit Removing the fluff, the question is:

When solving problem X by heart, how does the mind reaches the solution very fast? by 'running' an algorithm or 'accessing' a table


The 'fluffy' version:

Since I was a kid I was always amazed by mental arithmetic. Today I still am.

With time I got to realize that the best engineers, physicists, mathematicians around me seem to not only handle very fast and precise mental arithmetic but also fast and precise mental proportions and spatial relationships.

therefore my question is: are they solving these problems in a algorithm like fashion (but faster than most people) or are they getting these results from the back of their mind from some sort of "table", where they archived many results through the years and just retrieve them "without effort"?

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    $\begingroup$ Mental arithmetic is useless to a mathematician. $\endgroup$ – UserX Nov 29 '14 at 11:26
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    $\begingroup$ Some tables are useful, like the multiplication table, prime numbers, powers of primes and certain constants (but never more than three digits). $\endgroup$ – Frank Vel Nov 29 '14 at 11:43
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    $\begingroup$ If we are talking also about $\pm$ operations I remember my math teacher talking about his mom, who works at a cash desk and did all those "simple" arithmemtics very fast (doing it all day).. Personally I sometimes feel like I am taking a guess and then checking it. I think, the answer won't be much further from both is true. $\endgroup$ – quapka Nov 29 '14 at 11:48
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It's not really an answer to your question but there is a section (titled "Lucky Numbers") in the book "Surely You're Joking Mr Feynman" about how physicist Feynman mastered mental arithmetic under the tutelage of another physicist Hans Bethe. He describes remembering many values and trick methods, so it is a combination of both things you describe.

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  • $\begingroup$ thank you for the reference! $\endgroup$ – Draconar Nov 29 '14 at 12:31
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Like in many matters the more things you know the more become easily accessible (as you may deduce from Feynman's very funny book cited by Suzu Hirose : knowing just some logarithms helped him much!).

For example knowing the table of logarithms in base $10$ may help you greatly to get order of size of complicated multiplications, divisions and powers (or $n$-root).
Of course calculators (and smartphones) will very well do the job these days but it is always kind of ridiculous to see someone stretching the thing out when the task is to divide by $10$...
The same way people will ask "why do we need to learn integration techniques?" and so on since the computers do the job.

Anyway the answer given by Asimov "the Feeling of Power" is not that inappropriate... But there is a more immediate consideration : because it is fun and at all ages as you may see at he highest level here.

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As a complement to my comment;

enter image description here

Also, remember that the first computers were invented to make people's lives easier with arithmetic.

On the other hand, spatial relations are important, but practice makes perfect.

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  • $\begingroup$ I surely know that @UserX, but I see this tendency on very good "math heads" of dealing very fast with say.. numbers, proportions, relationships, etc. If they lack mental arithmetic skills, they sure have some of the others laying around $\endgroup$ – Draconar Nov 29 '14 at 11:40
  • $\begingroup$ @Draconar do you have a citation for that? $\endgroup$ – UserX Nov 29 '14 at 11:54
  • $\begingroup$ stripping the fluffly the question was "When solving problem X by heart, how does the mind reaches the solution very fast? by 'running' an algorithm or 'accessing' a table". $\endgroup$ – Draconar Nov 29 '14 at 12:43

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