[Jprogramming] Another Quora Problem

['Mike Day' via Programming]

At the risk of getting too chatty,  I'd point out that in the majority 
of Project
Euler problems*,  while you can (sometimes) explore the foothills of the 
challenge,
including example solutions provided,  with a brute-force approach, it 
is generally
necessary to find ways to avoid requiring too much of either or both 
memory and
compute time.

BTW,  I recall the story that Gauss answered the sum of integers problem 
in his
High School class,  by considering,  say,  the sequence 1 2 3 ... 20 21,
and its reverse,  21 20 ... 3 2 1,  showing that each pair of items from 
each sums
to 22,  so that the sum of the two sequences is 21 * 22...   No doubt 
apocryhal.

Cheers,

Mike

* see https://projecteuler.net

On 29/08/2017 09:04, Skip Cave wrote:
> There seems to be two basic approaches to this problem:
>
> 1. Generate the even numbers between 1 & 42, and add them up.
> 2. Use the formula for the sum of an arithmetic progression - e.g.:
>
> Sum=n(a+b)/2
> *n* = number of numbers in the sequence (here 21)
> *a* = the first number in the sequence (here 2)
>
> *b* = the last number in the sequence (here 42)
>
> Generally, I like to approach these kinds of Quora sequence problems using
> a "brute force" approach, and ignore simplifying formulas. In the days
> before modern computers, formulas were developed to make computations on
> sequences like this Quora problem more tractable, particularly when the
> number of terms got large.
>
> Today, with powerful modern computers and languages, it is often easier to
> simply generate a sequence in it's entirety, and then compute some property
> of that sequence such as it's limit, sum, etc. to get the final result.
> This is particularly true when you have at your fingertips a powerful
> matrix language such as APL or J.
>
> For me, the obvious approach to this problem is to simply generate the
> sequence of even integers and then sum them, instead of trying to find a
> formula that relates to that specific sequence (in any case, I didn't have
> a clue where to look for such formulas).
>
> So the plan was: Generate the numbers from 0 to 42, throw out the odd
> numbers (zero doesn't matter in this problem), and then add up what's left:
>
>      a =:i.43
>
>      +/(-.2|a)#a
>
> 462
>
>
> My main dissatisfaction was that I was sure that this could be done in a
> single line, but I didn't know how to get the integer sequence on both
> sides of the tally/copy verb, which i wanted to use as a binary selector
> mask.
>
> Yes, thanks to Raul and others, I now realize that I could have just
> multiplied the first 21 integers by 2 to get all the even integers and then
> sum them, but I wanted to learn how to get the same vector of integers in
> two places in the formula without having to assign that vector to a
> variable.
>
> Raul showed me how to do that, though he used equality & multiplication to
> generate the selector mask and make the selections.
>
>      +/(*0=2&|)i.43
>
> 462
>
>
> Raul's multiply by zero trick was cool, but my purist sensibilities still
> wanted to use selection of the even numbers as the most straightforward way
> to implement the sum of evens.
>
> However, Raul's approach did show me the way. I could use the copy operator
> to implement the selector mask, instead of multiplication
>
>     +/(#0=2&|)i.43
>
> 462
>
>
> That worked, So we should also be able to get the sum of the odd numbers:
>
>
>     +/(#2&|)i.43
>
> 441
>
>
> Yep. That worked too. So i should also be able use the NOT verb ".-" to
> invert the selection mask to get the sum of evens like I did in my original
> attempt:
>
>
>      +/(#-.2&|)i.43
>
> 43
>
>
> AAAAK!  What happened? I tried to negate the selection mask, and something
> went terribly wrong! Maybe I need to isolate the negation:
>
>
>      +/(#(-.2&|))i.43
>
> |length error
>
> | +/ (#(-.2&|))i.43
>
>
> Nope!  What am I doing wrong? How can I simply negate the selection mask,
> just like I did in my original explicit example?
>
>
>
> Skip Cave
> ----------------------------------------------------------------------
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