The One2Six Advantage-Play Project, Part 6: Short Physical-Card Returns Are Strongly Suppressed

The simulator preserved the average return time while reducing same-card returns within 20 cards by approximately 92%

Part 5 established the recurrence framework and confirmed that the experiment could recover the expected geometric distribution when the source was genuinely memoryless. Part 6 applies the same measurement to labelled physical cards moving through the six-deck One2Six-style source.

This produced the first result that materially changed the direction of the project.

Across one million draws, the average return time of a physical card was almost identical under the physical-IID baseline and the One2Six model:

Source	Mean cards between appearances
Physical IID	310.90
One2Six, recycle batch 20	310.95

The average looked exactly as a six-deck population would suggest. Each of the 312 physical cards continued to account for roughly one draw in every 312 over the long run.

The timing of those returns was far less ordinary.

Under physical IID, the same card returned with 20 or fewer other cards between appearances approximately 6.52% of the time. Under the One2Six model using recycle batches of 20, the corresponding rate was 0.53%.

That represents a reduction of approximately 92%.

The effect also remained when cards were returned individually. With a recycle batch of one, the same card returned within 20 cards only 1.58% of the time, approximately one-quarter of the physical-IID rate.

The mean was perfectly respectable and almost completely unhelpful. The return-time distribution showed that the machine was preserving long-run frequency while heavily suppressing the quickest physical-card returns.

What the Experiment Measured

The experiment tracks the recurrence of individual physical cards.

A six-deck game contains six separate copies of every rank-and-suit combination. Six cards may all be tens of spades, although they remain different physical objects moving through the mechanism.

The simulation assigns each card a stable identity. A target within the One2Six source might be represented as:

one2six-deck-0:T:spades

The corresponding target in the physical-IID baseline might be:

physical-iid-deck-0:T:spades

When that card appears, the experiment records how many other cards are dealt before the same physical object appears again.

This matters because the One2Six moves actual cards through a sequence of physical states. A dealt card leaves the active source, enters a recycle batch, passes through the feeder, reaches a carousel shelf, waits for ejection, joins the output buffer and eventually returns to the table.

A rank-level analysis would merge the six physical tens of spades into one category and lose that history. It could measure how long until another ten of spades appeared, but it could not distinguish the original card from one of the other five copies.

The recurrence experiment therefore required a null model containing 312 labelled physical cards.

The Six-Deck Physical-IID Null

The physical-IID source contains:

6 decks
312 labelled physical cards
independent selection from all 312 cards on every draw
no depletion
no discard rack
no feeder
no carousel
no output buffer

It does not represent casino dealing. It provides the clean memoryless comparison for the question being asked: what would the return-time distribution look like if every physical card had an equal and independent chance of selection on every draw?

For one specified card:

p = 1 / 312

Let (G) represent the number of other cards appearing between successive appearances of that target. Under physical IID:

P(G = g) = (1 - p)^g × p

The expected number of cards between appearances is:

E[G] = (1 - p) / p

     = 311

The probability of the same card returning with (g) or fewer cards between appearances is:

P(G <= g) = 1 - (1 - p)^(g + 1)

For 20 or fewer cards:

P(G <= 20)
    = 1 - (311 / 312)^21

    ≈ 0.0652

Approximately 6.52% of physical-IID same-card returns should therefore occur within that window.

Validating the Recurrence Measurement

Before applying the metric to the One2Six, I ran one million draws through the physical-IID source.

Two specified target cards were tracked separately, while recurrence intervals were also pooled across all 312 physical cards. Each individual card was expected to appear approximately:

1,000,000 / 312 = 3,205.13 times

The observed target counts were:

Target card	Observed appearances
Physical ten of spades	3,232
Physical five of spades	3,159

The pooled mean also closely matched the theoretical value:

Metric	Expected	Observed
Mean cards between appearances	311.00	310.90

More importantly, the measured distribution followed the geometric reference across the full range:

Cards between appearances	Observed probability	Theoretical probability
50 or fewer	0.1514	0.1510
100 or fewer	0.2771	0.2769
250 or fewer	0.5532	0.5533
500 or fewer	0.8001	0.7998
1,000 or fewer	0.9596	0.9598

The experiment recovered both the expected mean and the expected shape when the underlying process was memoryless. That gave me confidence that any later difference reflected the source rather than a badly defined recurrence metric.

The One2Six-Style Source

The source used for this experiment was the configurable six-deck mechanism introduced in Part 4. Its physical card path was:

dealt card
    -> held outside the source
    -> returned in an ordered recycle batch
    -> feeder
    -> carousel shelf
    -> shelf ejection
    -> back of the output buffer
    -> front-buffer draw

A recently dealt card cannot become immediately available on equal terms with every other card. It first has to complete that path.

The working configuration was:

Parameter	Value
Deck count	6
Carousel shelves	38
Shelf capacity	10
Output-buffer target	18
Refill threshold	8
Minimum preferred ejection occupancy	7
Ejection rule	Whole shelf
Shelf order	Last-in, first-out
Accepted-discard ingestion	Immediate

These values remain modelling assumptions. The manuals and patents support the broader mechanism family of individual feeding, compartment storage, group output and a front shoe. They do not establish every parameter in the table as a production One2Six setting.

The point of the configurable source is to keep those assumptions visible and allow later experiments to move them.

The First One2Six Recurrence Run

The initial experiment used:

draws = 1,000,000
deck_count = 6
recycle_batch_size = 20
seed = 42

This was a source-level experiment. It contained no blackjack hands, betting, strategy or settlement. Cards were drawn, held outside the source and returned in ordered batches of 20.

The mean return time again looked entirely normal:

Source	Mean cards between appearances
Physical IID	310.90
One2Six, recycle batch 20	310.95

The cumulative distribution revealed how differently those return intervals were allocated:

Cards between appearances	Physical-IID theoretical	One2Six, batch 20
20 or fewer	0.0652	0.0053
50 or fewer	0.1510	0.0529
100 or fewer	0.2769	0.1665
250 or fewer	0.5533	0.5228
500 or fewer	0.7998	0.8235
1,000 or fewer	0.9598	0.9765

At the 20-card threshold:

Physical IID:       6.52%
One2Six, batch 20:  0.53%

The One2Six model produced only around 8% of the physical-IID short-return rate.

The gap remained substantial at 100 cards. By 250 cards, the One2Six distribution had recovered much of the missing probability, and by 500 cards its cumulative return probability had moved above IID.

The machine had delayed the returns rather than removing them. Probability that would have occupied the immediate part of the distribution shifted into medium and later intervals.

Why the Mean Hid the Result

The model contains 312 conserved physical cards. Over a sufficiently long run, each card continues to contribute approximately one draw in every 312, provided the mechanism does not permanently favour or suppress particular cards.

That keeps the average recurrence interval close to 311 cards.

The mean alone says very little about how those intervals are distributed. Physical IID allows the same card to return on the very next draw because every selection comes independently from all 312 identities. The One2Six source forces a recently dealt card to pass back through the feeder, carousel and output buffer.

Those stages remove probability from the shortest intervals. The delayed cards still return later, allowing the long-run mean to remain almost unchanged.

This was the type of structure the project was designed to detect. A comparison based only on average card frequencies or average recurrence would have described the two sources as nearly identical.

The shape of the distribution showed that their short-term behaviour was very different.

Testing the Effect of Recycle-Batch Size

Returning cards in batches creates an additional delay. A card at the beginning of a recycle group remains outside the source while the rest of the group is accumulated.

The first result therefore needed to be separated from the effect of using batches of 20. I repeated the one-million-draw experiment with recycle batch sizes of 1, 5, 20, 52 and 100, while keeping the six-deck One2Six configuration unchanged.

Recycle batch	Mean between	20 or fewer	50 or fewer	100 or fewer	250 or fewer	500 or fewer	1,000 or fewer
1	310.93	0.0158	0.0697	0.1895	0.5313	0.8175	0.9730
5	310.94	0.0130	0.0664	0.1840	0.5288	0.8185	0.9735
20	310.95	0.0053	0.0529	0.1665	0.5228	0.8235	0.9765
52	310.94	0.0023	0.0295	0.1318	0.5057	0.8336	0.9815
100	310.96	0.0012	0.0171	0.0898	0.4731	0.8507	0.9884

Larger batches strengthened the delay, which was consistent with the physical structure of the experiment. The important result came from the batch-size-one run.

When every card was returned individually:

One2Six, batch 1:  1.58%
Physical IID:      6.52%

The One2Six rate was approximately 24% of the IID expectation, representing a reduction of around 76%.

Returning cards one at a time removed the waiting period required to accumulate a larger recycle group. The remaining suppression came from the mechanism itself. Each card still had to enter the feeder, reach a shelf, wait for shelf ejection, join the back of the output buffer and move forward until it could be dealt.

Recycle-batch size controlled the strength of the delay, while the feeder, carousel and buffer created the underlying effect.

What the Result Establishes

Within the current six-deck One2Six-style model:

• same-card recurrence differs materially from physical IID;
• returns within very short intervals are strongly suppressed;
• substantial suppression remains when cards are recycled individually;
• larger recycle batches increase the delay;
• probability removed from the shortest intervals appears later in the distribution;
• the mean recurrence interval remains close to the physical-IID expectation.

This is the first clear structural result from the machine model. Its physical state retains enough history to reshape the return-time distribution across one million draws.

The finding is more specific than saying that the machine appears streaky or that certain sequences look unusual. It measures one defined property against a calibrated 312-card null model and shows how the difference changes as recycle-batch size moves.

For me, this was the point where the simulator moved beyond being an interesting reconstruction exercise. It had produced a result strong enough to determine what the next experiment needed to be.

Why This Matters for Advantage Play

The current model creates a period in which recently dealt physical cards are less likely to return because they remain somewhere along the route back to the front of the output.

That temporary suppression changes the active population from which the next cards can emerge. If the recent cards contain an unusual concentration of low cards, ten-value cards or aces, their delayed return could make the short-term output conditionally richer or poorer in the corresponding categories.

Physical identity itself is unavailable to the player. A player cannot distinguish one physical ten of spades from the other five copies. Any practical signal therefore has to survive aggregation into visible categories such as:

• low cards;
• ten-value cards;
• aces;
• Hi-Lo values;
• recent discard composition.

Part 6 establishes a mechanical reason why those aggregate signals could exist. It does not yet show that the physical-card suppression remains visible after the identities are combined into ranks and blackjack values.

That became the next question for the project.

Limits of the Result

The experiment establishes behaviour inside the current model. The exact compartment count, buffer depth, insertion order and ejection rules of a production One2Six remain uncertain, and the simulator represents a documented mechanism family rather than proprietary production firmware.

The result also sits at the beginning of the advantage-play chain. Mechanical structure has to survive aggregation into visible information, predict future cards before a wager, change blackjack expected value materially and remain robust across assumptions, seeds and practical table conditions.

The recurrence finding clears the first of those stages. The model has measurable short-horizon memory, and the effect is large enough that it cannot be dismissed as minor sampling noise.

The remaining stages are harder because they require the hidden physical effect to survive contact with the limited information available to a player.

Relationship to Earlier Public Work

Physical casino shufflers have previously been analysed as probability-generating systems. Persi Diaconis, Jason Fulman and Susan Holmes examined casino shelf-shuffling machines and showed that their mixing behaviour could be studied through the underlying mechanism.

Their work concerns shelf-shuffler mixing rather than the continuous physical-card recurrence experiment performed here, although the broader principle is closely related: a physical machine generates a particular stochastic process and can be studied as such.

Stephen How’s public continuous-shuffler work examined recently inserted cards, buffer depth, discard timing and short-window counting. That work is closer to this project because it also treats the machine as a stateful process whose recent input may affect later output.

The measurement chain used here was:

312 labelled physical cards
    -> calibrated physical-IID null
    -> configurable One2Six-style carousel source
    -> pooled same-card return distribution
    -> recycle-batch sensitivity

I did not find this exact labelled-card recurrence comparison and batch-sensitivity experiment in the public material I reviewed. That does not mean the underlying idea has never occurred to anyone. It means the implementation, null model and reported result appear distinct from the public analyses I found.

Where the Project Stood After Part 6

The earlier parts had reconstructed the mechanism, built the simulator and calibrated the recurrence framework. Part 6 produced the first result strong enough to redirect the investigation.

At a recycle batch of 20:

Physical IID, return within 20:  6.52%
One2Six, return within 20:       0.53%

With cards returned individually:

Physical IID, return within 20:  6.52%
One2Six, return within 20:       1.58%

The delayed cards eventually returned, keeping the mean close to the six-deck expectation. Their timing had changed substantially.

The next stage was to determine whether that hidden physical memory survived when the cards were grouped into information visible at the table. Part 7 examines recent discard composition, the duration of any observable effect and whether the resulting signal reaches blackjack expected value.

References

• mathematical-ev/shufflemaster-simulation. Public repository containing the physical-IID source, One2Six-style source, recurrence experiments and recycle-batch sensitivity analysis described in this article.
• Persi Diaconis, Jason Fulman and Susan Holmes, Analysis of Casino Shelf Shuffling Machines. Relevant prior work showing that physical casino shuffling mechanisms can be analysed as probability-generating processes.
• *Stephen How, Counting CSM Blackjack (+EV).* Relevant public work on continuous-shuffler state, recently returned cards, discard timing and output-buffer effects.
• Dimitri P. Bertsekas and John N. Tsitsiklis, Introduction to Probability. Relevant to geometric waiting times, independent sampling and construction of the physical-IID recurrence baseline.