Electricians - Why "3-Way" Switches?

Nope, that doesn’t do it. Maybe I can explain it a little better.

Previously we had only used switch A to control the lights. B and C are in relatively distant parts of the basement. At some point in the weekend, the downstairs guests were using B and C to switch the lights on and off. By the end of the weekend, all the lights were off. Switch A was in the down (off) position. B was also down but C was up. Now, whatever combination of switches I flip, one switch is always in the opposite position.

Right now, for example, the three switches are in down/up/down position, and the lights are off. I need to find the sequence to flip that eventually ends up with all the switches in down position, with the lights off.

There is no such sequence. If the other switches in the basement are so remote, then it’s possible you were just misremembering how they used to be. That, or your guests opened up the walls and re-wired the switches. The switches only care about what positions they are each in, not what order they were flipped.