Twinbrix likes to gamble, and if you play and win, he will reward you with a Shaft Gate Operating Dial (Key Item). This key item is required to enter the level 60 ENM battles Pulling the Strings and Automaton Assault.
Twinbrix will roll a number between 2 and 100. Depending on how much gil you trade him will determine what the winning numbers will be. If you trade him 2,000 gil, you will win if he rolls between 2 and 10. If you trade more gil instead, the high win number will increase. This maxes out if you trade him 10,000 gil, with which you will win if he rolls between 2 and 50 (a roughly 50% chance to win).
If you trade him 2,716 gil. His normal dialogue changes and he says "2716 gil? Is that supposed have somes kinda meaning?", before stating you need to roll between 2 and 13 to win. Trading 2,716 gil on Lightsday will improve your chance to win (q.v. Pulling the Strings).
Warning!! If you have a Shaft 2716 Operating Lever (Key Item) in your possession when you teleport, you will lose it.