American roulette keeps 38 pockets on the wheel, and European roulette keeps 37. Labouchère does not double the chip. The player writes a short row of numbers, and the bet equals the sum of the two ends. A win crosses off those two numbers, and a loss writes the amount staked onto the end of the row.
An even chance is red or black, odd or even, 1–18 or 19–36. On the RoulCoin wheel there are 18 winning pockets out of 38. There are 20 losing ones: eighteen cells of the other color and two greens, 0 and 00. The payout equals the bet. The row comes out only with that payout: a closed cycle brings exactly the sum of the numbers the row was started from.
French roulette on a single zero sometimes returns half the stake if zero comes up. In the tables below there is no such return. Both 0 and 00 take the chip in full, and the whole amount lands at the end of the row, not half of it.
How the row 1, 2, 3, 4 closes
The sum of the starting row is 10. That is the aim of the cycle, not the profit of one spin. The first bet is 1+4 = 5. While there is a number in the row, the cycle is not finished. Below is one path of seven spins: the “result of the cycle” column is counted from zero, before the first bet.
| Spin | Row before the bet | Bet | Outcome | Result of the cycle |
|---|---|---|---|---|
| 1 | 1, 2, 3, 4 | 5 | loss | −5 |
| 2 | 1, 2, 3, 4, 5 | 6 | loss | −11 |
| 3 | 1, 2, 3, 4, 5, 6 | 7 | win | −4 |
| 4 | 2, 3, 4, 5 | 7 | win | +3 |
| 5 | 3, 4 | 7 | loss | −4 |
| 6 | 3, 4, 7 | 10 | win | +6 |
| 7 | 4 | 4 | win | +10 |
The check comes out. The losses took 5+6+7 = 18. The wins returned 7+7+10+4 = 28. The difference equals 10, the sum of the numbers written before the first spin. After the seventh spin the row is empty. The next cycle starts from a new row, not from a remainder.
There is also a short path. Two wins in a row stake 5 and another 5, cross off all four numbers, and also give +10. On 38 pockets such a pair comes up in (18/38)² of cases: 22.44%, about 1 run out of 4.5. On 37 pockets (18/37)² gives 23.67%, about 1 out of 4.2. That is the frequency of the short path only. A long row does not close by itself because of it.
Clean losses in American roulette
If the color does not come up even once, the bet grows by one unit per spin: 5, then 6, then 7. This is not doubling. Ten such losses on a wheel of 38 pockets have probability (20/38)¹⁰, about 0.16%, roughly 1 series out of 613. On 37 pockets (19/37)¹⁰ is about 0.13%, roughly 1 out of 784.
| Loss | Bet | Already on the table | Next bet |
|---|---|---|---|
| 1 | 5 | 5 | 6 |
| 2 | 6 | 11 | 7 |
| 3 | 7 | 18 | 8 |
| 4 | 8 | 26 | 9 |
| 5 | 9 | 35 | 10 |
| 6 | 10 | 45 | 11 |
| 7 | 11 | 56 | 12 |
| 8 | 12 | 68 | 13 |
| 9 | 13 | 81 | 14 |
| 10 | 14 | 95 | 15 |
Ten losses leave 95 on the table, and the eleventh bet equals 15. The same depth under doubling from a start of 1 is bets from 1 to 512 and a sum of 1023. With the row 1, 2, 3, 4, on the tenth loss it is the bankroll that hits the ceiling, not the size of one chip. At a maximum of 500 the table will still accept a bet of 15. In an account the ceiling can be different: it is checked before the first row.
The table’s share on every bet of the row
Plus 10 is obtained only by the cycle that was played out to an empty row. Each separate spin remains a bet on a color. Per 100 staked an American wheel takes 5.26 on average. A European wheel, where zero also takes the stake in full, takes 2.70. A return of half on zero lowers the loss to about 1.35, and on the RoulCoin wheel this rule is absent.
| Wheel | Losing pockets | Per 100 staked | How zero enters the row |
|---|---|---|---|
| American, 38 | 20 | 5.26 | 0 and 00 write on the whole bet |
| European, 37 | 19 | 2.70 | the single 0 writes on the whole bet |
| French, half on zero | 19 | about 1.35 | the tables above do not have this return |
The length of the row does not heal the share. A short one closes more often and brings a small plus. A long one less often requires one large bet, but a broken cycle takes the sum of all the numbers written on. Plus 10 in the example does not subtract 5.26 after the fact: those 5.26 sit inside every spin, including a winning one.
Plus 10 on an empty row is the sum of 1, 2, 3 and 4, not a new payout of a wheel of 38 pockets.
How not to mix up the row if you play online
For a player in England the ends of the row do not depend on the city. 18 winning pockets out of 38 and two green zeros are the same in any time zone. The row is better written before the spin, not assembled from memory: one missed figure changes both the bet and the aim of the cycle.
Playing online without signup is possible on a virtual balance. Guest chips do not turn into a withdrawal, and for checking the row that is enough: it shows how a loss stands at the end and how two ends disappear after a win. Live chips are connected separately, once the table ceiling and the length of the row are already written down.
- The starting row is short and written down before the bet. In the example the aim of the cycle equals 10.
- A win crosses off both ends. The one number left is the next bet.
- A loss writes on the whole bet, including a spin with 0 or 00.
- The cycle is stopped when the next bet does not fit into the bankroll taken in advance.
Why one end cannot be crossed off
If after a win only the left number is removed, the sum of the remaining row stops being the aim of the cycle. A win of 7 with the row 1, 2, 3, 4, 5, 6 removes both 1 and 6. What remains is 2, 3, 4, 5, and the next bet is 7 again. One crossed-off end leaves an extra number, and an empty row no longer gives exactly +10.
What is staked if one number is left in the row?
That number is the bet. In the table the seventh spin stakes 4, because only 4 is left in the row. A win empties the row and brings the result of the cycle to +10.
After a win, can one end be crossed off?
No. A win of 7 with the row 1, 2, 3, 4, 5, 6 removes 1 and 6 together. If only 1 is removed, the remaining numbers no longer agree with the aim of +10.
Does zero write half the stake onto the row?
On a wheel of 38 pockets, no. 0 and 00 take the whole stake, and it enters the row in full. Half the stake is a different rule, on a single zero, and these tables do not have it.
Do ten losses in a row already require a bet above 100?
No. The tenth bet in a clean series equals 14, by that moment 95 is on the table, and the next bet is 15. The written-down bankroll runs out before the ceiling of a single chip.
Does the same row work on a Dozen?
No. A Dozen pays 2 to 1 and wins on 12 pockets out of 38. Crossing off two ends is calculated for a payout equal to the bet, not for a double one.
The row is easy to take for a schedule by which the table gives back the sum of the starting numbers. Neighboring playing strategies are put together differently: the bet grows either after a loss, or only after a win. Before moving the ends onto live chips, the row is worth closing once on paper by the table above.