how do you make a java applet?

I didn't look at your code extensively, but from the image you've provided of the GPQ solver, I'd have to say your methods could use refining. All it does is prevent you from trying combos that are known to be wrong, and it's not guaranteed to prevent failure. I'd suggest you go back to the drawing board and have the program make suggestions on what combos you should try next. You can do this by having the computer pre-compute probabilities and stuff for chance of success. It's sorta what people programming chess AIs do; they assign "points" to certain spots, and spots with the highest points are where the AIs are most likely to move.

In terms of actual GPQing, there's the "finish in 5 moves or less" method which it is recommended a computer should do, my method which I developed (that thread is hella old lmao, and I see that you've visited it), and the "try four of each item" method. Assuming perfect executiong of each method, the last one has a small chance of failure but probably gets you the fastest times out of all three, while the first one is infuriatingly hard to do with just people.