Published experiments, with receipts.
Fixed seeds, stated stopping rules, downloadable tables, and results that can be regenerated. The build log remains below; the studies are the durable record.
The question
Every gambler chases the same carrot: the system that finally beats the wheel. Against a fair wheel, betting systems can't create edge, only shape variance, and that's most of what this platform instruments today: drawdown depth, time-to-ruin, terminal-bankroll spread. A real wheel is also a physical object, and physical wheels have been beaten before. The studies separate those questions: fair-wheel betting systems, statistical bias detection, and physical prediction are not interchangeable claims.
Studies
Martingale: 100,000 sessions
A fixed-seed experiment measuring how often a $5 Martingale reaches a $100 target before bankroll or table limits break the progression.
Study 02Gambler’s ruin validation
A 140,000-trial check of the gambler’s ruin formula against deterministic simulation on European roulette even-money odds.
Study 03Wheel-bias sample size
Power calculations for detecting a roulette pocket that lands 0.25 to 2 percentage points more often than the fair 1/37 baseline.
Study 04European vs. American risk
Paired-seed simulation of 20,000 sessions per wheel showing how a second zero changes terminal bankroll, drawdown, and ruin risk.
Study 05RNG design and validation
Frequency, serial-correlation, repeatability, and multi-seed batch checks for the exact two-stage seeded generator used by 0xRoulette.
Study 06Same-seed strategy comparison
A paired comparison of Flat, Martingale, Fibonacci, D’Alembert, and Paroli under identical European-wheel outcome sequences.
Study 07Drawdown and ruin methodology
A worked methodology for peak-to-trough drawdown, finite-horizon ruin estimates, and Wilson confidence intervals using 25,000 fixed-seed sessions.
Build log
Product changes remain here so a reader can distinguish a published result from a feature release.
Housekeeping, not a feature: the running text on Home, Research, Physics, About, and Support was hitting the container's edge on tablet widths, so the narrow-content column went from 840px to 900px. Small, but it's the kind of thing that quietly signals "nobody proofread this on a real screen," and that's not the register we want.
The other two are link hygiene. TrainingSharks (our sister project, same engine-first build, pointed at a game that's actually beatable with skill) now gets a reciprocal footer link; it already linked here. And the session-summary footnote on /play, which previously only stated the fixed theoretical return, now closes with the same nudge toward the reading list that the marketing pages already carry. Anyone who lands straight on /play and never scrolls a marketing page was missing the one honest ask this project makes. Now they see it exactly once, in the same dry parenthetical voice as the rest of the stat panel.
Eight small additions, one goal: put the house edge on screen in real time instead of leaving it implied. The live table read-out (derived straight from the engine's own payout resolver, not a re-derivation) shows coverage, true win probability, and expected value per spin for whatever's on the felt right now: drop $25 on a dozen and a street and watch the EV go red before you spin, not after. The hot/cold heat map does the companion job for the physical half of the thesis: a confidence-scaled tint per pocket, with an in-app note that on this fair RNG a hot run is variance, not signal, the same distinction Jagger's scratch-marked wheel and Ethier's 1982 sample-size math draw on the Physics and Research pages, now sitting on the board itself instead of only in the prose.
The rest is scaffolding for the same honesty: a shareable run seed (?seed= in the URL, one deterministic sequence per seed, manual and auto spins both routed through it) so a result can be handed to someone else and reproduced exactly; a ten-step guided tour that replaces the old bank auto-open for first-timers; a session summary with net P&L, peak, max drawdown, and streaks for the spins since page load; and saved bet templates so testing a layout twice doesn't mean re-clicking it twice. None of it moves the edge. All of it makes the edge harder to look away from.
Shipped the first pass of the metrics that make variance legible instead of a feeling: maximum drawdown, the time-to-ruin distribution, and terminal-bankroll spread across a backtest. These are the numbers that separate "feels safe" from "is safe." Martingale posts a gorgeous win rate and a fat left tail; the drawdown metric is where that tail finally shows up on screen instead of ambushing you on spin 200.
The math underneath is old and settled. The gambler's-ruin problem, worked out in full in Feller's Introduction to Probability Theory, Vol. 1, gives the exact probability that a bankroll hits zero before it reaches a target when every bet carries negative expectation. We don't just trust it; the deterministic engine simulates the same setup and we check the empirical curve against the closed form. They agree. That agreement is the whole reason to keep the engine reproducible.
The autopilot personas now run four textbook progressions side by side (Martingale, D'Alembert, Fibonacci, and Labouchère), each with its own live P&L. None of them beats the wheel; that was never the experiment. The experiment is watching how differently they lose. Martingale bets the farm to claw back a nickel. D'Alembert bleeds slowly and smoothly. Labouchère hides its risk inside a list you have to read twice. Same −2.7% expectation, four completely different variance signatures.
This is the point Richard Epstein makes in The Theory of Gambling and Statistical Logic, still the classic catalog of the field: every betting system is a re-parameterization of one fixed edge. You can move the risk around (front-load it, smear it out, disguise it), but the integral under the curve doesn't move. Seeing that on four synchronized equity curves is more convincing than any paragraph, including this one.
Started the real design work on wheel-bias modeling, the feature that turns "hunt for real-world physics" from a slogan into something you can do here. Before writing a line of it, we did the statistics homework: how many spins does it actually take to prove a wheel is biased? Ethier's 1982 paper in the Journal of the American Statistical Association, "Testing for Favorable Numbers on a Roulette Wheel," works out the sample sizes, and they are humbling. Detecting a single over-represented pocket at any honest confidence level takes thousands to tens of thousands of logged spins. Joseph Jagger's clerks in 1873 weren't being thorough for sport; that's simply the sample size the math demands.
The other half of the record isn't bias at all: it's physical prediction. Small and Tse showed in Chaos (2012) that clocking a real wheel's initial position and velocity yields roughly an 18% edge with a hidden clicker: the modern, peer-reviewed descendant of the Eudaemons' shoe computer. Both halves, persistent bias and single-spin prediction, are getting instrumented here. Neither is a promise that you'll win. Both are a promise that it'll be a fair fight against real math.
The simulator existed before the brand did. This week we gave it a name, 0xRoulette, and a face: a light/dark themed site, a research notebook (this page), and the honest business scaffolding every public tool needs. Privacy, terms, affiliate disclosure, consent-gated analytics. Nothing about the engine changed; we just stopped hiding it behind a single unlabeled page.
The thesis is unchanged: roulette is a fixed negative-EV game, and over the long run no strategy has closed the gap on the house edge. We take the carrot seriously enough to test it honestly, and to report exactly where, and how, it slips out of reach. What we study is variance, the shape of the chase, not a phantom edge we promise you'll find.
The backtest kernel runs in a dedicated Web Worker with a 30-second watchdog, so a strategy looping over thousands of spins never blocks the UI and a wedged run can't poison the next. Two independent seeded PRNGs drive each run: a pocket RNG for spin outcomes, and a separate strategy RNG exposed to Lua as ctx.rng(), seeded by (seed ^ 0x9e3779b9) so a strategy's own randomness can't accidentally correlate with the wheel.
Reproducibility is the feature that makes results comparable across strategy iterations. math.random is nil'd inside the sandbox specifically to force the deterministic path.
The math, worked
None of this is proprietary: it's the arithmetic every number on this site rests on, written out so you can check us. If any of it is wrong, that's a bug; tell us.
Where −2.7% comes from
A single-number ("straight up") bet on a European wheel pays 35:1 but wins with probability 1/37. The expected value of a $1 bet is (1/37)(+35) + (36/37)(−1) = −1/37 ≈ −0.0270, a 2.70% loss per unit wagered, and the same −1/37 falls out of every bet the European layout offers, straight-up or even-money, because the single green zero is the only asymmetry. Add the second green pocket and the American wheel doubles it: −2/38 ≈ −0.0526, a 5.26% edge. That constant is what no ordering of bet sizes can touch. It is the wall.
Gambler's ruin: the price of chasing a target
Say you flat-bet even money, winning each round with probability p = 18/37 and losing with q = 19/37. Starting with i units and quitting at N, the probability of going broke first is ((q/p)^i − (q/p)^N) / (1 − (q/p)^N). With q/p ≈ 1.056 that ratio compounds against you fast: the further your target sits above your bankroll, the more overwhelmingly likely ruin becomes before you ever reach it. This is the closed form our time-to-ruin simulation reproduces, and the reason "just double until you win" is a statement about when you go broke, not whether.
Why Kelly says stake nothing
The Kelly criterion (Kelly, 1956) sizes a bet to maximize long-run log-growth: f* = (bp − q) / b. On any bet against a fair wheel bp − q is exactly the house edge (negative), so the growth-optimal stake is f* < 0. The math's advice for beating a fair wheel with a betting system is, literally, to hold the other side or bet zero. That's not defeatism; it's the theorem. It's also why the only positive-Kelly play in this whole history came from people measuring the wheel, not the odds: see Physics.
How many spins to prove a wheel is crooked
Bias-hunting is a hypothesis test: is some pocket's true frequency above the fair 1/37? A pocket over-appearing by even a full percentage point hides easily inside ordinary variance over a few hundred spins. Ethier (1982) works out the observation counts rigorously; the practical upshot is that credible bias detection lives in the thousands-to-tens-of-thousands of spins, which is exactly why documented cases in the record involved months of hand-logging. When wheel-bias modeling lands here, the simulator will expose that sample-size wall directly. The current planning calculation is published in the wheel-bias study.
References & further reading
The primary sources behind the claims on this page and on Physics. Where a book is easy to buy, it's on our reading list: those are affiliate links that help keep this free.
- Ethier, S. N. (1982). Testing for Favorable Numbers on a Roulette Wheel. JASA 77(379), 660–665. (The sample sizes real bias detection actually requires.)
- Small, M. & Tse, C. K. (2012). Predicting the outcome of roulette. Chaos 22(3), 033150. (Peer-reviewed physical prediction: a measured ~18% edge from timing the wheel.)
- Kelly, J. L. (1956). A New Interpretation of Information Rate. Bell System Technical Journal 35(4), 917–926. (The bet-sizing criterion that says stake nothing against a negative edge.)
- Thorp, E. O. (1969). Optimal Gambling Systems for Favorable Games. Review of the International Statistical Institute 37(3), 273–293. (The formal line between beatable and unbeatable games.)
- Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1 (3rd ed.). Wiley. (Gambler's ruin, in full, in Chapter XIV.)
- Ethier, S. N. (2010). The Doctrine of Chances: Probabilistic Aspects of Gambling. Springer. (The modern reference text for the probability of casino games.)
- Epstein, R. A. (2009). The Theory of Gambling and Statistical Logic (rev. ed.). Academic Press. (The classic catalog of betting systems and why each one only reshapes variance.)
- Bass, T. A. (1985). The Eudaemonic Pie. Houghton Mifflin. (The shoe-computer story, told from the inside.)
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