Given the following contrived problem:

Print out the sum of the numbers from 1 to 100.

The average programmer would probably see this as something to be written with a loop, a nice O(n) problem, and might code it in perl as follows:

#!/usr/bin/perl -w

use strict;

my $n = 100;
my $sum = 0;

$sum += $_ for 1 .. $n;

print "$sum\n";
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Give the same problem to a mathematician, and might be seen as:

Sum(1..n) = n(n+1)/2
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however it may ultimately be coded -- a nice O(1) problem.

The programmer will implement the algorithm provided in the most efficient way possible, hopefully, whereas the mathematician is the one who should provide the most efficient algorithm.

If these two work together, we might get a program like this:

#!/usr/bin/perl -w

use strict;

my $n = 100;
my $sum = $n * ( $n + 1 ) / 2;

print "$sum\n";
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which runs in essentially the same amount of time for any number, n, rather than taking longer and longer to run -- see first program, above.

Many times I've seen people say, "Let the programmer solve it all...he'll make it go fast!" But what about the other disciplines? By involving the mathematician, her algorithms are implemented by the programmer to produce even faster code.

How many of your projects incorporate the disciplines necessary for the realization of the most efficient solution possible?

Pondering,
-v

Update:

Kudos to St. grinder for catchin' a typo: a few missing $ signs in the last example.