Interesting problems that can be solved with algorithms I need to write a blog post on a problem that can be solved algorithmically. I need to do two things:

1) I need to decide what the naive brute force method would be and touch on it's time and Space complexities.

2) I need to come up with a more elegant/ideal method to solve the problem, then discuss tradeoffs with the ideal method's time and space big O

Does anyone have ideas for an interesting topic I can cover? I'm not looking for the answers, just the topic to cover

Here is an example of something I could write about: https://medium.com/@hannahsquier/two-algorithms-for-solving-vigneres-cipher-in-ruby-77bf0b37507a#.go43z5vas
Depending on the level of sophistication of your intended audience, a simple canonical example is testing whether an input number $n$ is prime. 

A brute force approach is to test, for all $d$ in the range $2,\dotsc n-1$ whether $d$ divides $n$. If you find a $d$ that divides $n$ you can stop and assert that $n$ isn't prime; if you don't find any such $d$, you can assert that $n$ is prime. That'll take about $n$ divisions if $n$ is prime. 

An improvement is to try just those $d\le\sqrt{n}$ (rounding up), since it's easy to see that if $n$ has any divisors, there must be one in that range. This is a big improvement, since it reduces the number of divisions from $n$ to $\sqrt{n}$, more or less.

If you want to expand this, there are even faster ways to test for primality and if your audience is sophisticated enough, you might want to try one or more of them.