Personally I would start with an ltrace of the program with any arbitrary set of arguments. I'd then use the strings command and guess from that what some of the hidden argument literals might be. (Let's assume, for the moment, that the professor hasn't encrypted or obfuscated the strings and that they appear in the binary as literals). Then try again with one or two (or the requisite number, if number).
If you're lucky the program was compiled and provided to you without running strip. In that case you might have the symbol table to help. Then you could try single stepping through the program (read the gdb manuals). It might be tedious but there are ways to set a breakpoint and tell the debugger to run through some function call (such as any from the standard libraries) and stop upon return. Doing this repeatedly (identify where it's calling into standard or external libraries, set a breakpoint for the next instruction after the return, let gdb run the process through the call, and then inspect what the code is doing besides that.
Coupled with the ltrace it should be fairly easy to see the sequencing of the strcmp() (or similar) calls. As you see the string against which your input is being compared you can break out of the whole process and re-invoke the gdb and the program with that one argument, trace through 'til the next one and so on. Or you might learn some more advanced gdb tricks and actually modify your argument vector and restart main() from scratch.
It actually sounds like fun and I might have my wife whip up a simple binary for me to try this on. It might also create a little program to generate binaries of this sort. I'm thinking of a little #INCLUDE in the sources which provides the "passphrase" of arguments, and a make file that selects three to five words from /usr/dict/words, generates that #INCLUDE file from a template, then compiles the binary using that sequence.