American roulette pays one unit to one for a color and takes the stake on 20 pockets out of 38. European roulette on the same even chance takes the stake on 19 pockets out of 37. After a loss, grand martingale does not stake a merely doubled chip: one more unit is added to the double.
The row starts at 1. After that the bet equals the previous one doubled, plus 1: 3, 7, 15, 31, 63, 127, 255, 511. A win on step n closes the series at plus n, not at plus 1. That is the whole difference from ordinary doubling, where any successful step returns only one unit on top of the series.
Zero and double zero get no separate rule. Both green pockets are the same loss of the color as the other cell, and the next step is counted by the same formula.
How the step grows in American roulette
The column “already lost” is the sum of the bets before this step. “Result” is what remains if this step wins and the series closes. The start is one unit everywhere.
| Step | Bet | Already lost | Result if the step won |
|---|---|---|---|
| 1 | 1 | 0 | +1 |
| 2 | 3 | 1 | +2 |
| 3 | 7 | 4 | +3 |
| 4 | 15 | 11 | +4 |
| 5 | 31 | 26 | +5 |
| 6 | 63 | 57 | +6 |
| 7 | 127 | 120 | +7 |
| 8 | 255 | 247 | +8 |
| 9 | 511 | 502 | +9 |
A check of the fourth step: 1+3+7 = 11 have been lost, and a bet of 15 returns them and leaves +4. The eighth step, after seven losses, stakes 255 with 247 already lost and, if it wins, gives +8. The ninth bet is 511. Ordinary doubling at the same depth of eight losses takes 255; here 502 is already on the table.
Where the bankroll ends before the ceiling
Eight losses in a row have the same probability as under ordinary doubling: (20/38)⁸, about 0.59%, roughly 1 series out of 170. On 37 pockets (19/37)⁸ is about 0.48%, roughly 1 out of 207. What costs more is not the frequency, but the sum.
| Steps already lost | On the table | Next bet | What a bankroll of 500 does not allow |
|---|---|---|---|
| 4 | 26 | 31 | the bet goes through |
| 5 | 57 | 63 | the bet goes through |
| 6 | 120 | 127 | the bet goes through |
| 7 | 247 | 255 | 253 remains, short by 2 |
| 8 | 502 | 511 | both the bankroll and the bet of 511 are already behind |
A table ceiling of 500 would still have accepted the bet of 255 itself, and would not accept the bet of 511. A bankroll of 500 units, after minus 247, no longer assembles the bet of 255: 253 is in hand. The figures 500 and 100 below are for teaching. In the account the limit may be different; it is checked before the first unit, not on the seventh loss.
The share of American roulette does not pay for the extra unit
Every step remains a bet on a color. Per 100 staked an American wheel takes 5.26, and a European wheel takes 2.70. The addition of “plus one” increases both the bet and the plus of a successful step, but not the number of winning pockets. French roulette with half the stake on zero does not enter this row: 0 and 00 take the chip in full.
| Rule | Eight losses | On the table | Plus, if the next step won |
|---|---|---|---|
| Ordinary doubling from 1 | 1…128 | 255 | +1 |
| Grand martingale from 1 | 1, 3, 7 … 255 | 502 | +8 |
Plus 8 against plus 1 looks more generous only for as long as the eighth bet has been placed and has won. The price of entering this step is the 502 already lying on the table plus the bet of 255 itself.
A win on step n closes the series at plus n. For that plus the row asks a sum that outruns ordinary doubling by almost twice.
How not to mix up the step
For a player in England, 0 and 00 do not become a reason to add a second extra unit. The formula is one: double the previous bet and add 1. A time zone does not rearrange the pockets.
- After a win the row starts again from 1. The winning plus is not carried into the next ladder.
- The series is stopped when the next bet does not fit into the bankroll set aside in advance.
- A bet of 255 at a ceiling of 500 may be allowed by the table and still not fit into the remainder of the bankroll.
- The color is not changed inside the row: changing the field does not reduce the 247 already lost.
Why the plus on paper is larger than the plus of the evening
One closed series gives a small whole number, from +1 to +n. One series broken on the seventh step takes 247 and does not leave a bet with which this minus closes, if 500 was in hand. Several short pluses are not obliged to catch up with one such break.
How does a step of 15 differ from an ordinary bet of 8?
Ordinary doubling on the fourth step stakes 8 and, on a win, gives +1 with 7 already lost. Here the fourth step is 15 with 11 already lost, and a win gives +4.
Does a bankroll of 500 reach the bet of 255?
No, if seven losses have already taken 247. 253 remains, and the step asks 255. A table ceiling of 500 does not by itself forbid this bet yet.
At a maximum of 100, which bet is the last?
The bet of 63 is less than 100, and the next one, 127, is already greater. But after five losses 57 is on the table, and out of a bankroll of 100, 43 remains for the sixth step. The bet of 63 cannot be assembled from that remainder.
After zero, are two units added, not one?
No. 0 and 00 are an ordinary loss. The next bet is counted as the previous one doubled plus 1, with no second addition for the green pocket.
The extra unit in the formula makes a successful step more noticeable and a break more expensive. Neighboring playing strategies diverge in the sum on the same streak of eight losses: 255 under ordinary doubling and 502 here. The row is worth writing out before the first chip, while “plus n” does not yet hide the price of entry.