D'Alembert System Calculator

Defaults: Base Unit $10, Current Units 5. d’Alembert staking on even-money chances raises the wager by one unit after a loss and lowers it by one unit after a win, never below one unit. This page does not play the up-and-down. It prices the triangular exposure if you have climbed from 1 unit to `units` on a pure losing run, and it names the current stake. Max Exposure = base × units × (units + 1) / 2 = 10 × 5 × 6 / 2 = $150.00. Current Unit Bet = base × units = $50.00. That $150 is 10+20+30+40+50, the money already posted if every step from one unit through five was a loss.

A $50 even-money win at this height returns $50 profit on the spin, which does not fill a $150 hole. You would then drop to 4 units ($40) and keep grinding. The equilibrium fantasy — that wins and losses should balance, so you end up with one unit profit per resolved pair — is why the system exists and why it fails. Losses are slightly more common than wins on a 37-pocket even-money bet (19 versus 18), so the walk drifts up the unit ladder more than it drifts down. The triangular pile is the cash that drift can demand.

Common mistakes: treating Current Units as “five wins,” starting the triangle at zero, and believing $150 is a worst-case over a long session rather than the specific 1-through-5 losing path. A real d’Alembert path mixes wins and losses; your live exposure is path-dependent. Next, set units to 1: exposure $10, bet $10. Set units to 10: exposure 10×10×11/2 = $550, current bet $100. Compare that $550 with Fibonacci’s slower early curve and Martingale’s $310 after only five doubles: d’Alembert is the mildest of the three at short depth and still unbounded if you keep losing.

Do not read $150 as a prediction. It is the triangular number T_5 times the unit. On European even money you still face a 2.70 percent house edge on every dollar of turnover, including the $50 current bet. Try units 3 versus 7 to see how T_n = n(n+1)/2 accelerates once the unit count leaves the single digits.

Jean le Rond d’Alembert’s name got glued to an even-money progression that adds one unit after a loss and subtracts one after a win. The Enlightenment mathematician wrote about equilibrium in physics and probability; gamblers heard “equilibrium” and decided red and black should even out, therefore a ladder that buys more when behind and less when ahead would harvest the balance. That is the equilibrium fallacy. Independent spins have no memory. This calculator exists to show the cash shape of the ladder — triangular numbers — not to validate the philosophy.

Who uses it: roulette-system historians, players who find Martingale psychologically violent and want the “mild” alternative, and teachers demonstrating that mild plus negative edge is still negative. Monte Carlo and other 19th-century houses listed d’Alembert beside Labouchère in the same brochure of progressions. None of those brochures included a proof of an edge, because there isn’t one. The house edge on even money is 1/37 or 2/38 depending on zeros, independent of unit count.

What it does not do: simulate the +1/−1 walk, compute the probability of reaching 5 units, include table limits, or apply la partage. There is no wheel selector. Max Exposure assumes a monotone climb from 1 to `units`. If you reached 5 units after a mix of results, you did not necessarily spend $150. The current bet of $50 is still correct for “I am on 5 units now” regardless of path; the $150 is the pure-loss-path total.

Limitations. Unbounded in theory: nothing stops Current Units except a max-bet sign or an empty wallet. In practice the system feels safe because stakes grow linearly, not geometrically. Linear growth still ruins you if the session is long and the edge is negative — it just takes more spins than Martingale. The calculator will not tell you to step up to $50. It will tell you that five units at $10 is a $50 stake and a $150 monotone-loss pile. Educational, not a forecast, not financial advice.

Triangular numbers, not equilibrium, are how d’Alembert exposure is calculated on a monotone losing climb. A base of 10 and five units produce the arithmetic series 1 + 2 + 3 + 4 + 5. The fifth triangular number is T_5 = 5 × 6 / 2 = 15. Exposure is that triangle times the unit: 15 × 10 = 150 dollars, which is also 10 + 20 + 30 + 40 + 50. Current bet at five units is 10 × 5 = 50 dollars. That 150 is the money already posted if every step from one unit through five was a loss and the stake rose by one unit after each miss, as the system prescribes.

Jean le Rond d’Alembert (1717–1783) contributed to the Encyclopédie and to probability debates, including the notorious claim that after heads on a fair coin the next toss is no longer one-half. The betting ladder that bears his name is a popular attribution rather than a casino memoir he signed. Gamblers heard “equilibrium” in his mechanics and decided that red and black should even out, therefore a ladder that buys more when behind and less when ahead would harvest the balance. That is the d’Alembert equilibrium fallacy. Independent Bernoulli trials with win probability 18 / 37 have no memory. Wins are never due after losses, so raising the stake is not justified by maturity of chances.

Expectation on a European even-money unit is minus 1 / 37 of a unit. Raising to five units makes the expectation minus 5 / 37 of a unit on that spin — larger in dollars, identical in edge. If the plus-one / minus-one walk is modelled as a biased random walk, the drift is toward higher units because losses are slightly more frequent than wins. Recurrence to one unit is not guaranteed in a finite session; ruin or a table maximum is. A 50-dollar even-money win at this height returns 50 of profit on the spin and does not fill a 150-dollar hole. The walker then drops to four units and continues. Milder than doubling is not the same as profitable.

Worked comparisons belong beside the triangle. At one unit, exposure is 10 and the current bet is 10. At ten units, T_10 = 55, exposure 550, current bet 100. Linear growth feels safe and is still unbounded. T_20 = 210 units would be 2100 dollars of monotone-loss exposure at a 10-dollar base, with a current bet of 200 that many tables will refuse. The chance of a five-loss even-money run is (19 / 37)^5 ≈ 3.57 percent European, the same event a doubling lecture prices; d’Alembert does not reduce that chance. House edge is untouched by the name of an Enlightenment mathematician.

Assumptions: monotone losing climb from 1 to 5, even-money payout not required for the triangular arithmetic, ability to post the current 50. A mixed live path that happened to sit on five units did not necessarily spend 150. The 50-dollar current bet is still correct for “five units now” regardless of path; the 150 is the pure-loss-path total. Base 10, units 5, T_5 = 15, exposure 150, current bet 50 are that bookkeeping. They are not a claim that the ladder returns to equilibrium, and they are not a forecast of the next even-money result.