Finder of biased coins.
I don't see the point beyond use as a coding exercise, unless you are a researcher in the field or your job is to develop such applications (e.g. you are a developer for Intel MKL).In production, yes, As student, doing it from scratch is better. And it teaches you algebraic manipulation..
Now, use LU in combination with ADI solver for option pricer (btw there is hardly any ADI code on the web).
And even as a coding exercise, you are going to have to be more explicit with why it's better for students. What does doing it even teach? I've done it (implemented LU) myself before, and all I really remember was that I had headaches doing the bookkeeping to make sure my indices were right.