The One2Six Advantage-Play Project, Part 7: Discard History Predicts the Next Deal and Nearly Erases the House Edge

11 July

Written By Paradis

A player-observable fading count survived held-out seeds and moved the strongest state from a 0.46% loss to effectively break-even

Part 6 established that the One2Six-style model retains physical memory. Recently dealt cards were strongly suppressed at short return intervals, even though the average time between appearances remained close to the six-deck physical-IID expectation.

That finding answered an important mechanical question, although physical-card identity is hidden from a player. Nobody sitting at the table can distinguish one physical ten of spades from the other five copies circulating through a six-deck game. For the effect to matter in practice, it had to survive aggregation into information a player could see before betting.

Part 7 follows that path from hidden machine state to visible discard history and then into blackjack expected value.

The visible discard rack predicted the composition of the next deal. Batches that had already been returned to the machine continued to carry a weaker predictive effect for roughly another 100 cards. A simple fading count built from that history survived testing on untouched simulation seeds and predicted low cards, ten-value cards and aces separately.

The economic result was the most important finding in the project so far. Across the full validation sample, the unconditional One2Six game lost approximately 0.462% of the initial wager. In the strongest predicted high-card-rich state, which occurred in just under 10% of rounds, the point estimate improved to:

-0.002%

By point estimate, the observable signal recovered approximately 99.6% of the house edge.

The confidence interval around that state still crossed zero, so the result does not yet establish a reliably positive game. It does show that ordinary table-visible information predicted the next deal strongly enough to move the estimated edge from a conventional blackjack loss to approximately break-even in the most favourable score band.

The project had now connected the mechanical structure inside the simulator with information a player could observe and use before placing the next wager.

From Physical Memory to Observable Information

Part 6 tracked labelled physical cards and found that the One2Six-style source heavily suppressed very short same-card returns. A card that had just appeared could not immediately rejoin the full six-deck population on equal terms. It first had to be collected, returned to the machine, fed into a shelf, released into the output buffer and moved forward until it reached the dealer.

That established a clear form of mechanical memory.

The practical problem was that a player observes ranks and suits rather than internal physical identities such as:

one2six-deck-0:T:spades

The player sees a ten of spades and can record that information, but cannot know which physical copy appeared or where it later sits inside the carousel.

The next stage therefore examined whether the suppression of individual cards remained visible after the cards were grouped into categories available to a player. Those categories included low cards, ten-value cards, aces, Hi-Lo values and the composition of recent discard batches.

The result was clear enough to justify taking the analysis further. The model’s hidden physical state left a measurable trace in the visible history of the game.

Validating the Complete Blackjack Game

Conditional profit results are easy to produce if the game around them is wrong, so I first ran a large comparison of the complete blackjack engine under physical IID and the One2Six source.

The rule profile used for these experiments included:

six physical decks;
no dealer hole card during the initial deal;
blackjack paying 3:2;
dealer standing on hard and soft 17;
doubling on 9, 10 or 11;
doubling after a split;
one card dealt to each split ace;
no ordinary resplitting;
original-wager-only treatment when the dealer later makes blackjack after a split or double;
an initial burn card;
return of the preceding discard rack after the next initial deal when a shuffling device is used.

The stand-on-soft-17 rule required a correction from an older public description of the game. I updated the engine to the current formal rule profile before running the large validation experiment. There is little value in defending an earlier implementation once the governing rule is available.

The player used one active box, a flat wager and the same fixed strategy under both card sources. The strategy could see the player cards, dealer upcard and legal actions. It had no access to the source type, physical card identities, carousel shelves, feeder contents, output-buffer state, previous discard history or internal telemetry.

Only the card source changed.

The comparison used five independent seeds and one million rounds per source.

Metric	Physical IID	One2Six
Mean player edge	-0.5489%	-0.5675%
Player blackjack rate	4.7467%	4.7566%
Double rate	8.6753%	8.6711%
Initial equal-value pair rate	14.8050%	14.4892%
Split rate	2.6558%	2.5253%
Win rate	43.3474%	43.3403%
Loss rate	47.7809%	47.8612%
Push rate	8.8717%	8.7985%

The two sources produced extremely similar long-run blackjack results. The paired monetary difference was only -0.0186 percentage points, its confidence interval crossed zero comfortably, and the direction changed between seeds.

Blackjack rates, doubles, wins, losses, pushes and overall monetary performance all landed in credible ranges. The large simulations also completed without a physical-card invariant failure or fallback ejection.

This mattered because the later score was intended to identify favourable states within an otherwise ordinary-looking blackjack game. A source producing obviously distorted unconditional outcomes would have made any conditional result much harder to interpret.

The Lower Pair Rate Was a Useful Validation Result

The most visible difference between the two sources was the rate of equal-value starting pairs. Physical IID produced more pairs and consequently more split opportunities than the One2Six source.

My first reaction was to inspect the dealing and strategy logic because an unexpected pair-rate difference could easily indicate a bug. Direct instrumentation showed that the difference closely matched the mathematics of the two sources.

Under physical IID, the first and second player cards are independent selections with replacement. When the ten-value ranks are grouped into one blackjack value, the theoretical equal-value pair probability is:

P(equal-value pair) = 25 / 169

                    ≈ 14.7929%

The observed physical-IID rate was:

14.8050%

The One2Six contains a finite circulating population. Once the first card is dealt, one matching physical card is unavailable for the second draw. The corresponding finite six-deck probability is approximately:

14.5189%

The observed One2Six rate was:

14.4892%

The difference between the sources closely matched the known difference between independent sampling with replacement and drawing from a finite physical population.

This gave the validation more depth than a collection of broadly plausible headline rates. The source reproduced a subtle physical-card effect whose direction and approximate size could be calculated independently.

It also provided a useful reminder that an unexpected result deserves investigation before being classified as either a discovery or an error. In this case, the result was evidence that the One2Six source behaved like a circulating pack of physical cards rather than a memoryless symbol generator.

Streaks Were Not the Signal

I had suspected that the dependence inside the machine might alter player winning and losing streaks. The first summaries showed nearly identical means and percentiles, but those statistics alone could have missed differences in tail weight, continuation probability or serial dependence.

I therefore ran a dedicated streak-shape audit over two million measured rounds.

The streak definition was monetary. A positive total result for the active box counted as a win, a negative result counted as a loss and a zero result counted as a push. Split hands were combined before the round was classified, while pushes neither counted nor interrupted an active streak.

The audit examined complete run-length distributions, geometric benchmarks, survival curves, continuation probabilities, transition matrices, tail ratios and outcome autocorrelation.

Metric	Physical IID	One2Six
Mean win streak	1.9014	1.9051
Geometric implied win mean	1.9042	1.9063
Mean loss streak	2.1028	2.1020
Geometric implied loss mean	2.1060	2.1033
Win-streak 95th percentile	5	5
Loss-streak 95th percentile	5	5

The empirical streak distributions followed the geometric benchmarks closely. Continuation probabilities remained approximately constant, and the autocorrelation of outcomes was effectively zero.

I also tested whether streak sign and length improved held-out monetary prediction after the card-composition score was included. They made the prediction slightly worse.

That was enough to close the line of investigation. The machine’s physical dependence remained clear at card level, but the process of dealing complete rounds, applying strategy and settling outcomes washed it out almost completely when the game was reduced to wins, pushes and losses.

There was no useful reason to keep streaks in the candidate advantage signal merely because they sounded intuitively relevant.

The Visible Discard Rack Predicted the Next Deal

The next experiment examined information available before the following wager.

Under the procedure being modelled, the preceding round’s discard rack remains outside the machine until after the next initial deal. Every card visible in that rack is therefore unavailable during that deal.

This creates a direct compositional effect. If the rack contains an unusually large number of low cards, those low physical cards have temporarily left the population capable of appearing. The remaining population should become relatively richer in ten-value cards and aces. A rack containing an unusually large number of high cards should shift the available population in the opposite direction.

For every current rack, I recorded:

rack size;
Hi-Lo count;
low-card count;
neutral-card count;
ten-value count;
ace count.

The simulator then cloned the exact pre-deal state and inspected the next 15 cards while leaving the current rack outside the source. Fifteen cards cover the full initial deal for seven player boxes and the dealer upcard.

Physical IID provided the negative control. Because each IID draw is independent, the rack should carry no information about future cards.

The estimated slopes were:

Predicted response	Physical IID slope	One2Six slope
Hi-Lo composition	0.637	1.132
Low-card frequency	0.481	0.987
Ten-value frequency	0.883	1.260
Ace frequency	-0.165	1.165

Every physical-IID confidence interval included zero. Every One2Six interval excluded zero, and the responses moved in the direction implied by temporary removal.

A low-heavy rack predicted fewer low cards, more ten-value cards, more aces and a more high-card-rich composition over the next 15 draws.

This provided the first direct connection between the hidden physical mechanism and information available to a player. The player did not need access to the carousel or output buffer and did not need to identify individual physical cards. The visible discard rack contained enough information on its own.

The Machine Had a Measurable Memory Horizon

The current rack provides the strongest information because its cards remain outside the machine. Earlier racks have already been returned and are moving through the feeder, carousel and output buffer.

The next experiment measured how long the composition of those returned batches continued to affect later output.

Cards dealt after batch return	Hi-Lo response	Low response	Ten-value response	Ace response
1 to 15	0.744	0.748	0.777	0.807
16 to 50	0.394	0.381	0.483	0.448
51 to 100	0.201	0.199	0.187	0.244
101 to 250	0.002	0.005	0.011	0.002
251 to 500	-0.001	0.006	0.009	0.005
501 to 1,000	-0.024	-0.022	-0.017	0.011

The information was strongest immediately after return, weakened materially through the first 50 cards, remained detectable through roughly 100 cards and had effectively disappeared beyond that range.

The practical horizon was therefore much shorter than the physical-card recurrence effect measured in Part 6. Exact recurrence remained distorted over several hundred cards and extended towards 1,000. Once physical identities were aggregated into ranks and blackjack values, the observable response faded more quickly:

strong through 15 cards
moderate through 50 cards
weaker but detectable through 100 cards
negligible beyond 100 cards

I had also considered whether a returned high-card-rich batch might produce a later reversal when those cards eventually emerged from the carousel in a group. The data did not support a predictable reversal wave.

The usable effect came from temporary suppression followed by gradual dilution. The cards did eventually return, although the model provided no evidence that they became usefully “due” at a later point.

That was a less dramatic mechanism than a delayed burst, but it was also cleaner and easier to reconcile with the observed decay.

Building a Fading Observable Count

The measured memory horizon suggested a simple score that weighted discard cohorts according to their age.

Discard cohort	Frozen weight
Current visible rack	1.00
Returned 1 to 15 cards ago	0.75
Returned 16 to 50 cards ago	0.40
Returned 51 to 100 cards ago	0.20
Older than 100 cards	0.00

The coefficients were deliberately rounded. I wanted a score that reflected the broad empirical decay without optimising every weight against the development data.

The score used only information a player could observe or estimate:

card composition;
rack membership;
approximate age of previously returned cohorts.

It did not use:

physical card identifiers;
carousel shelf assignments;
feeder state;
output-buffer contents;
machine telemetry.

Development used seeds 42 through 46. Once the weights were chosen, they were frozen.

Validation used untouched seeds 47 through 51.

The score continued to predict the future card stream:

Predicted outcome	One2Six held-out slope	Seed-level interval
Hi-Lo	0.733	0.416 to 1.050
Low cards	0.865	0.715 to 1.015
Neutral cards	0.868	0.620 to 1.117
Ten-value cards	0.695	0.396 to 0.994
Aces	0.791	0.484 to 1.099

Every One2Six slope was positive across all five held-out seeds. Every paired One2Six-minus-IID difference was also positive across all five.

The one-box initial deal showed an even stronger response:

Initial-deal outcome	One2Six slope
Hi-Lo	1.023
Low cards	1.087
Ten-value cards	1.093
Aces	1.138

The validation mattered as much as the size of the estimates. The weights were fixed before the new seeds were examined, so the result could not be attributed to repeatedly adjusting the score until the same data agreed with it.

A count based entirely on visible discard history predicted the composition of the actual next deal.

The Signal Reached Money

Composition prediction provided the necessary bridge from the physical mechanism to blackjack. The next experiment tested whether the same score also ordered the expected value of the game.

Score bands were defined using development-score distributions only. Monetary outcomes were not used to choose the boundaries.

The fixed strategy was then played continuously across new validation seeds.

Score state	Frequency	Player edge
Strong high-rich	9.853%	-0.002%
Moderate high-rich	19.960%	-0.356%
Neutral	40.306%	-0.459%
Moderate low-rich	20.215%	-0.721%
Strong low-rich	9.666%	-0.673%

The unconditional One2Six edge was approximately:

-0.462%

The strongest high-card-rich state produced:

-0.002%

The improvement was 0.460 percentage points, recovering approximately:

0.460 / 0.462 = 99.6%

of the house edge by point estimate.

The score also sorted card composition in the expected direction. Comparing the strongest high-rich and low-rich states:

Player-card metric	Strong high-rich	Strong low-rich
Low-card frequency	37.17%	39.90%
Ten-value frequency	31.73%	29.53%
Ace frequency	8.01%	7.54%
Player blackjack frequency	5.111%	4.423%

The score was identifying genuinely different card environments, and the monetary outcome moved with those environments.

The strongest band did not clear the formal positive-EV gate. Its confidence interval extended across zero, and four of the ten individual seed estimates were positive. The point estimate was essentially break-even, while the uncertainty still allowed both a small player edge and a small house edge.

That limits the wording of the conclusion, although it does not make the result minor. A score using only observable discard history moved the game from its ordinary negative expectation to the neighbourhood of zero in roughly one round out of ten.

Negative Controls Strengthened the Result

Physical IID did not reproduce the same ordered compositional response. Its monetary slope was consistent with zero, and permutation-placebo tests also produced results consistent with zero.

One direct comparison was particularly informative. The improvement from the neutral state to the moderate high-rich state was:

0.476 percentage points more favourable
under One2Six than under physical IID

The seed-level interval was:

0.044 to 0.907 percentage points

That interval excluded zero.

The One2Six contrast considered by itself remained uncertain, but the difference between the two card sources was clearer. This supports the interpretation that the ordering came from the stateful mechanism rather than from dividing ordinary blackjack variance into several groups and selecting the most attractive one.

The IID controls were not perfectly silent across every secondary cell. With enough score bands, positions and action comparisons, isolated false positives appeared. That was expected and reinforced the need for frozen features, held-out seeds, direct source comparisons, permutation placebos and minimum support requirements.

Exploratory tables will usually contain an impressive number somewhere. The work lies in determining whether it remains impressive after the experiment stops helping it.

The Broad Strategy Search Did Not Produce a New Chart

Once the strongest state approached break-even, I tested whether composition-aware playing deviations could recover the remaining fraction of a percentage point.

A counterfactual action experiment collected 200,000 natural player decision states. Every legal action was branched from the same hidden game and source state, producing more than 407,000 action branches across development and validation seeds.

The composition state was updated at the actual decision point. By that stage, the previous rack had been returned after the initial deal, while the player cards and dealer upcard had become newly exposed and temporarily unavailable. Low-card, ten-value and ace richness were tracked separately.

No playing deviation passed the complete discovery and held-out validation gates.

This indicated that the fixed strategy remained robust across most of the compositional changes created by the machine. Any valid deviations were likely to occur around a small number of close threshold decisions rather than requiring a replacement strategy chart.

One exploratory lead remained:

player hard 16 against dealer ten
low-poor state
stand rather than hit

estimated improvement: 0.1241 initial wagers
interval: 0.0666 to 0.1836
matching states: 137

The estimated difference was large, but only 137 states matched the condition. It failed the support gate and remains a targeted hypothesis rather than a strategy recommendation.

The broader result points towards selective entry as the more promising source of value. Choosing when to play may capture more of the signal than making widespread changes to basic strategy after entering the hand.

What the Project Has Shown

Within the current six-deck One2Six-style model:

Finding	Result
Physical-card recurrence	Strongly non-IID at short horizons
Recycle-batch sensitivity	Short-return suppression survived
Complete blackjack game	Produced credible long-run outcomes
Finite-population pair rate	Matched the expected physical-card effect
Monetary streaks	Added no predictive value
Current discard rack	Predicted next-deal composition
Returned-batch history	Retained useful information for roughly 100 cards
Frozen fading score	Survived held-out validation seeds
Ten-value and ace prediction	Remained significant separately
Conditional expected value	Strongest state reached approximately break-even
Broad strategy deviations	None validated

The remaining gaps are equally important. No score state has yet produced a seed-level confidence interval entirely above zero. No wager spread, multi-box entry policy or live-machine test has been completed, and the production machine may differ materially from the current model.

Those limitations define the next stages of the project. They do not remove the evidence already accumulated.

The model retains physical memory. Some of that memory is visible in the discard history. The observable state predicts the future card composition and ranks the expected value of the game.

That is a substantive advantage-play finding, even though the final positive-EV threshold remains open.

What Appears to Be New

There is existing public work on physical shufflers and continuous-shuffler state.

Diaconis, Fulman and Holmes showed that casino shuffling machines can be studied as mathematical probability processes. Stephen How examined a continuous-shuffler model in which buffer depth, recently inserted cards and discard timing could create a next-hand counting effect.

This project follows the same broad principle of treating the shuffler as a stateful physical system. The complete evidence chain developed here is:

312 labelled physical cards
    -> physical-IID recurrence null
    -> configurable One2Six-style carousel model
    -> same-card recurrence analysis
    -> recycle-batch sensitivity
    -> complete blackjack validation
    -> visible discard-rack response
    -> returned-batch memory kernel
    -> frozen fading count
    -> held-out composition validation
    -> conditional blackjack profitability
    -> counterfactual action testing
    -> streak and placebo controls

I have not found a public reproducible analysis carrying that full sequence from labelled-card recurrence through held-out economic validation. The broad concepts have predecessors, while the implementation and combined evidence chain appear distinct in the public material I reviewed.

The central finding can be stated directly:

Under the current model, observable discard history predicts the next deal strongly enough to recover almost the entire blackjack house edge in the most favourable score band.

The Next Experiment

The strongest decile reached approximately break-even, but averaging the best 10% of states may conceal a smaller and more favourable extreme tail.

The next experiment therefore uses pre-specified thresholds covering:

best 20%;
best 10%;
best 5%;
best 2.5%;
best 1%;
corresponding low-rich tails;
the neutral centre.

The score cut points are defined without examining monetary outcomes and then applied to new validation seeds under both One2Six and physical IID.

The primary test is:

Does any pre-specified high-rich tail produce player expected value above zero with a seed-level confidence interval entirely above zero?

A successful result would point towards selective entry as the most direct practical policy. The player would reduce or avoid exposure during ordinary and low-rich states and participate only when the observable score crossed the validated threshold.

Bet spreading, additional boxes and playing deviations come later. The immediate task is to determine whether any observable state is reliably positive before optimising the amount placed at risk.

If the extreme tail does not cross zero, the next version of the score will replace the broad Hi-Lo-style weights with blackjack-specific effects of card removal and test the revised signal on fresh seeds.

The project has reached a point where both outcomes would be useful. A positive tail would establish a candidate entry policy. A failure would define how much information remains missing from the current score.

Reproducibility

The implementation, assumptions, tests and experiment runners are available in the public repository under GPL-3.0-or-later.

At this stage, the codebase contained 391 passing tests, with Ruff, formatting, mypy and compilation checks also passing.

Those checks validate the internal consistency of the software rather than the production-machine assumptions. Their value is that the model, calculations and reported results can be inspected and reproduced rather than accepted as a description of an inaccessible black box.

Final Thought

Part 6 showed that the model remembers physical cards. Part 7 established that part of this memory remains visible to the player.

The current rack predicted the next deal. Earlier returned batches retained a fading influence for roughly 100 cards. A frozen score survived untouched seeds, predicted ten-value cards and aces separately, and ordered the game economically.

The strongest observable state moved the estimated edge from:

-0.462%

to:

-0.002%

The result does not yet prove that a production One2Six can be beaten, and it does not clear the statistical threshold required to claim a positive game within the model.

It does establish a complete path from the reconstructed mechanism through player-observable information to a material change in expected value.

The next question is whether the most extreme high-card-rich states cross zero reliably.

References

mathematical-ev/shufflemaster-simulation. Public repository containing the physical-card sources, observable-state experiments, memory-horizon estimation, held-out score validation, conditional-profitability analysis and action-branching experiments described in this article.
Current Casino Blackjack Game Rules. Formal rule source used for the current game profile, including dealer behaviour, payouts, splitting, doubling and shuffling-device procedure.
Washington State Gambling Commission One2Six Shuffler Approval Memorandum. Relevant public regulatory material describing One2Six multi-deck operation, one-card feeding, random slot assignment and its rotating internal wheel.
Persi Diaconis, Jason Fulman and Susan Holmes, Analysis of Casino Shelf Shuffling Machines. Relevant prior work on analysing physical casino shuffling mechanisms as mathematical probability processes.
Stephen How, Counting CSM Blackjack (+EV). Relevant public work on recent-card windows, output-buffer depth, discard timing and state-dependent continuous-shuffler counting.

Paradis