21+3

1The rules, as data

A flat 9:1 on the three-card poker hand made of your first two cards and the dealer's upcard.

21+3 settles on the three-card poker hand formed by your two cards and the dealer's upcard. The version priced here pays a flat 9 to 1 on any flush, straight, three of a kind, or straight flush, and loses otherwise. The host game is six-deck blackjack with the dealer hitting soft 17. The bet resolves on the deal, so the blackjack rules barely touch it. What matters is the composition of the shoe feeding those three cards.

Suits matter here, which makes this bet unusual among the ones on this site. Blackjack itself never cares about suits, so my card model collapses them, and settling this bet runs on the raw (rank, suit) stream the shoe already carries underneath. Tiered paytables sold under names like Xtreme pay by category, run near a 13% house edge, and price completely differently. Nothing on this page transfers to them.

2The published-figure gate

The exact calculator matches the published house edge and, independently, the published perfect-play ceiling.

The fresh-shoe house edge computes exactly from the six-deck composition: 3.2386%. Eliot Jacobson's published analysis of the same game quotes 3.239%. His study also ran one hundred million shoes of computer-perfect play at a cut of 260 cards and reported a ceiling of 0.2748 units per 100 hands, and my exact ceiling at the nearest measured cut reads 0.269 to 0.276 units per 100 rounds. Two pipelines with no shared code landing inside each other's noise is the strongest kind of check I can run.

Calibration ran in the other direction too. Over three million rounds at the deep cut, realized results regressed on the pre-deal prediction give a slope of 1.034 ± 0.071 across bins spanning −13% to +9%, so the closed-form calculator prices depleted shoes correctly end to end.

3The oracle

This bet gets the full oracle: an exact EV formula at any shoe composition, evaluated before every deal.

The three-card category probabilities are polynomials in the per-(rank, suit) counts of the remaining shoe, so the exact EV has a closed form and evaluates before every round. The ceiling, betting exactly when that number is positive, reads +0.115 units per 100 rounds at a 0.75 cut and +0.276 at 0.85. The window at the deep cut is 7.1% of rounds at a mean of +3.88% per unit.

The signal lives late. The fraction of positive rounds runs 3.6% with three decks left, 12.5% at two, 25.1% at one, and 28.8% at half a deck, where the mean positive edge reaches +7.5%. Everything about operating this bet follows from that depth profile.

4The state dataset

Six million recorded rounds. The side-bet signal is nearly orthogonal to the blackjack count, and the main game charges rent.

Each recorded round banks the exact pre-deal EV next to the hi-lo true count and the depth. The correlation between the two signals is −0.07 to −0.09, so 21+3 windows and blackjack windows almost never collide. A counter running a hi-lo main game can stack this bet on top, and the pairing even replaces this bet's biggest cost with a positive leg.

That cost needs naming early because every later number carries it. The side bet requires a live blackjack wager, and basic strategy on this game runs −0.64% per round. Sitting through 100 rounds to catch seven windows pays the main-bet toll 100 times. The toll, and the two ways around it, shape the whole ledger.

5The fit, and the count

Suits carry 71% of the signal, ranks about 19%, and their interaction nothing. The value is a second moment, so the count is a sum of squares.

The category identities evaluate on fractional compositions, so each round's EV excess over a balanced shoe at the same depth decomposes exactly into a suit term, a rank term, and their interaction. Variance shares at the standard cut: suit 70 to 72%, rank 17 to 21%, interaction under 0.2%. The straight-flush cross term a 9:1 paytable could in principle reward measures dead. A computer betting on the suit-plus-rank sum recovers 99.8% of the ceiling.

The suit term has a structure no linear count can carry. A flush wants any one suit rich, and richness in spades pays the same as richness in hearts, so the value is symmetric in the four suits and quadratic in their surpluses. This is the site's clean example of the escalation clause in step five: the residual isn't a missing tag, it's a missing square. The human statistic that matches the structure is the sum over suits of the squared surplus, and its threshold curve comes out analytically, by bisection on the closed-form EV, with nothing fit to simulation.

Ranks add a little. The best static linear rank count on top of the four suit tallies buys about six more points of capture, at the price of a second 13-tag running count. The remaining fifth of the ceiling needs the full quadratic rank term, and that is computer territory.

The simple card in the box below shares its bookkeeping with Jacobson's published system, which tracks the spread between the fattest and thinnest suit, true-counted, and which he reports at 0.1777 units per 100 hands, about 65% of the ceiling. His article gives the count's shape and its return and stops short of printing the betting threshold, so the card below bets on the richest suit's surplus instead, using the threshold curve I derived analytically. Its certified capture is 47.3% at the deep cut, quad-Q reads 78.3% on the same shoes, and his published figure sits between them.

6Certified live

Every rule scored in true exact EV, with analytic parameters and nothing fit to the data it was scored on.

Each candidate card replays over two million rounds per cut with its bets scored against the exact calculator, and every threshold comes from bisection on the closed form, untouched by the simulation being scored. Quad-Q captures 74.2% of the ceiling at the 0.75 cut and 78.3% at 0.85, which lands within a point of the exact ceiling of its entire family, meaning any rule that sees only the four suit totals. The squared-surplus shape gives up nothing its inputs contain.

The intuitive alternatives fall well short. Betting when any one suit runs rich enough keeps 39 to 47% depending on the cut. A single-suit specialist keeps about 10%, which is where the old line about suit counting being worthless comes from. The published spread count sits at 65%.

7The ledger

The toll decides the mode. Seated play barely survives at the deep cut, and back-counting is the operating point.

Playing every round charges −0.64% of the main bet, every round. At the site's standard 0.75 cut that toll swallows the edge: seated play at a $15 main and $100 side is breakeven at best, and even back-counting clears only +0.083 units per 100. This page's scenario therefore runs at a 0.85 cut, and the deviation from the site standard is itself the finding. The bet works where the house deals deep, and moving the cut from 0.75 to 0.85 multiplies the net edge by roughly 2.5.

Back-counting, standing behind the table and betting only on trigger rounds, shrinks the toll to the trigger rate. Where a table caps the side bet at the main wager, matching both bets on trigger rounds keeps about 85% of the value, since a trigger round's window edge clears the main game's cost with room to spare.

8The verdict

Beatable, the first positive call I made, and priced like work.

The edge is real, certified, and pays about like a legitimate hi-lo counting operation while demanding a deeper cut and a $37k bankroll behind a $100 unit. The road is long too, with N0 near 120,000 observed rounds. The signal stacks cleanly on a hi-lo main game, so one seat can run both counts at once. And a maximum side bet that appears only late in the shoe is a pattern a floor can read, which matters more here than it would elsewhere, since suit-counting against this bet is already in the published literature.

§What this verdict depends on

• Flat 9:1 paytable on all four categories. The tiered versions run near a 13% house edge with different category weights, and nothing here transfers.
• Penetration is the kill condition: 0.75 to 0.85 is 2.5× the net edge, and below 0.75 only back-counting stays positive, thinly.
• Six decks. Every threshold on the cards is six-deck-derived.
• No continuous shuffling machine, and mid-shoe entry allowed for the back-counting mode.
• The tracker assumes all dealt cards become visible, including the dealer's hole card. A minor idealization at these stakes.
• The dollar figures count observed rounds, so table pace and seat availability set the hourly rate, and a full table shows fewer rounds per hour.

Last updated 2026-08-10.