Good for you! According to the latest message on the Resistance HQ page, "Please note that using any kind of external helper program or cheat program to solve this puzzle is against the rules and will likely get you FROZEN! So do the right thing and solve the puzzle yourself! (Discussions with friends and the use of reference charts is perfectly okay.) -- The Neopets Team"
Brute force is the only acceptable method. I guess if you magically enter the correct answer on the first try, they will ding you. Bummer -- I had worked out a way to code it in JavaScript today at work and was looking forward to writing the code tonight.
Oh well, back to my spreadsheet and brute force guessing. 
-Figure out what each combination of 2 symbols gets you. For instance, if you add B + Z, you get V. You don't need to calculate them for three and four.
-Find a symbol that shows up the most in your result, then use the information from the first part to figure out every possible combination of symbols that could create your target symbol in a column. The number of symbols used is the number of modifiers plus the starting line, so 4 symbols in a combination for the last 10 puzzles.
-For one symbol, cross off the combinations that do not include the starting symbol in that position (as they can't be made), then eliminate it from the remaining ones. You should now have a bunch of combinations of three symbols that, added to the one at the top, will form the desired symbol at the bottom.
-Write all possible combinations of modifiers that will give you that result. You don't have to write every one out; you can put them in columns.
-Repeat the above two steps for every instance the desired symbol occurs in the result. Ideally, you will wind up with overlapping sets of combinations. Find the combinations that are common to every set, and one of them should be the answer.
-While you do this, occasionally check to see if the combinations you're using actually add up to the desired symbols so that you don't wind up wasting a lot of time.
I suspect that the program written and banned earlier used a system like this, but with more symbols and combinations. You can call this method "targeted brute force."
Sadly, this method still takes as long as two hours (I spent an hour on a half at one that was near the end, but I can pretty much guarantee two hours will see a person through the whole cycle).
If they did close it, that would be really cruel.

And yes, I have an issue with TNT not letting people use calculators...how do they distinguish between someone who has written it themselves and someone using someone else's calculator? Anyway the way I see it, as long as you convert the symbol values to numerical values appropriately and have some way to run a check on all the combinations of 3 from the modifiers, you can find the right ones.
I had to complete the second puzzle before the door to the third puzzle opened. I didn't try going to the link directly...
I think right now they are checking to see if anyone accesses the text page with your puzzle combinations on it.