Linear Programming for board games?

I am a beginner to Linear Programming and am doing a course in it. So far
we've covered things like the simplex method and its dual, etc.

We have to come up with our own project, and I am hoping to use linear
programming to play a board game or some sort (e.g. othello, Connect-4,
checkers, or even tic-tac-toe). I have used neural networks and
evolutionary algorithms to play such games in the past with great success
(they could beat me), and hope to produce something similiar by using linear
programming.

So, is this possible? Doable for a beginner like me (assuming I'm willing
to do a bit of reading)? If so, what are some good on-line references?

I've been searching the net and found stuff on game theory and Nash
equilibria, but all examples I have seen have been for very simple games
(e.g. prisoner's dilemma).

Assuming I have a reasonable evaluation function to "score" how good a
particular board layout is, can I use something like Nash equilibria to look
a few moves ahead and pick a reasonable move? Or if not Nash equilibria,
what else can I use in the Linear Programming domain? I'm not after
building an unbeatable opponent... An OK Connect-4 player (for example),
that takes 5 minutes per move would still be acceptable, just as long as I
use Linear Programming for the project and show that I understand what is
going on. I can even use Linear Programming software to do all the
calculations for me, rather than doing it all by hand.

I get fairly easily confused by all the Greek maths and such that I am
seeing. So, if someone can point me in the right direction to a fairly
simple tutorial which explains what I am after, I would be eternally
grateful.

Thanks in advance,
Bull.