How the Titanic system works
Pick a dozen and bet 1 unit. After each loss, move to the next bet of the sequence: 1, 1, 2, 3, 4, 6. A dozen pays 2 to 1, so a win at any step covers the losses before it and ends the cycle with 1 or 2 units of profit.
After a win, start again at 1 unit on the dozen that just won. After six losses in a row, start again at 1 unit on the last dozen out.
Nine cycles out of ten, and the tenth
Six losses in a row on a dozen happen in 9.52% of cycles: (25/37) to the power of 6. So 90.5% of cycles win 1 or 2 units, and 9.5% lose 17. The rare loss erases about ten wins, and the house edge remains 2.70% of every bet.
Over whole sessions the picture is less rosy than over cycles. In the tester, with 1,000 to start, 5 per unit and 100 spins, 45% of sessions end in profit, none runs out of money, and the median session loses 15.