Blackjack, six decks, 20-10 pay table (suited pair 20, any other pair 10) · side bet
Pair Square
a side bet that your first two cards match in rank, sold under a half-dozen names and paying more when they match in suit too
Taking any single card out of a fresh shoe moves this bet by exactly zero, so no running count can read it. What it answers to is how bunched the leftover ranks are, one tally captures 91% of that, and the tally costs thirteen exact rank counts and a deep cut before it pays anything.
Every published system for this bet is a running count, and a running count cannot see it at all. That is a fact about the bet: pull any card out of a fresh shoe and the expectation doesn't move, to the last digit, and a balanced count has exactly zero covariance with the bet's value at every depth. Eliot Jacobson's card fires at a true count of +10 or −10, in both directions, which is the shape a system takes when it is reaching for something a count was never built to hold.
The bet answers to bunching. Pairs come more often when the survivors pile into a few ranks and less often when they spread out, and which ranks pile up doesn't matter. That quantity turns out to be one running number: add, for each card that appears, how many of its rank have already gone by. The tally and the depth together give the bet's value, and betting on it takes 90.2% of what a machine reading all fifty-two rank-and-suit counts would take.
The cut decides whether it pays. At the site's standard three-quarter shoe the tally clears $3.10 per hundred rounds after the seat, on a bankroll near $426,500, which is nobody's game. One deck deeper the same card clears $16.30 on $113,400, though N0 is about 464,000 rounds, the longest road on this site. That is the version priced here, and it asks for no blackjack count on top: the main bet stays flat on basic strategy and pays the toll, so the thirteen rank counts are the only thing being carried.
1The bet
Your first two cards, paid if they share a rank, paid more if they share a suit.
You post the bet before the deal, and it wins if your first two cards match in rank. Most tables pay extra when the two match in suit as well, and the ladders vary far more than the marketing does. The name changes by property: Pair Square, Bet the Set, Any Pair, and Perfect Pairs, which splits the unsuited win into same-colour and mixed-colour rungs.
The pay table decides almost everything here, so it gets named up front. I lead with six decks paying 20 for a suited pair and 10 for any other pair, because at 2.57% it is the cheapest version published and leaves the most room. Jacobson picked the same table as the most vulnerable, coming at it from a different direction.
Deck count matters more than it does on most side bets. A suited pair needs two identical cards, which one deck cannot produce at all, so every extra deck makes the top rung likelier and the bet cheaper. The 25-15-5 Perfect Pairs ladder runs 2.17% at eight decks and 4.18% at six.
| pay table | decks | suited pair | other pair | house edge |
|---|---|---|---|---|
| 20-10 | 6 | 20 | 10 | 2.57% |
| 12-12 | 6 | 12 | 12 | 3.86% |
| 15-10 | 6 | 15 | 10 | 10.61% |
| 25-8 | 8 | 25 | 8 | 4.10% |
| Perfect Pairs 25-15-5 | 8 | 25 | 15 / 5 | 2.17% |
| Perfect Pairs 25-12-6 | 8 | 25 | 12 / 6 | 4.10% |
2Checking against the published numbers
Eight published house edges, all reproduced by counting combinations, with no simulation involved.
against the published record
✓house edge 2.57% vs 2.57% published, six decks 20-10Wizard of Odds✓all four Pair Square ladders reproduced by enumerationWizard of Odds✓Perfect Pairs return −0.021686 vs −0.021687 published, eight decks 25-15-5Wizard of Odds✓most vulnerable at six decks with the 20-10 table, reached independentlyEliot Jacobson, Advanced Advantage Play★the pair tally is unpublished, and keeps 90.2% of the ceiling where Jacobson's published card keeps 20.6as far as I can findTwo cards settle this bet, which makes its value at any shoe composition a finite sum. Nothing here needs estimating, and that makes the gate unusually sharp. I counted the ordered pairs of undealt cards falling into each rung, divided, and compared against every pay table Wizard of Odds publishes for the Pair Square and Perfect Pairs families.
All eight agree. The tightest check is the one figure published to six places, the 25-15-5 ladder at eight decks, where the published return is −0.021687 and counting gives −0.021686.
3The ceiling
Exact at every composition, because the bet is small enough to count out in full.
Because the value is a finite sum, the perfect-information ceiling needs no estimating either. At any point in the shoe I can price the bet exactly from what remains, then ask how a player who somehow knew that number would do.
At three quarters penetration on the 20-10 table, that player bets 6.53% of rounds at an average of +2.16%, worth 0.1407 units per hundred hands. Everything below is measured against it.
4Where the value hides
Penetration buys variance and nothing else, so waiting for the end of the shoe is worth exactly zero on its own.
I walked 4,000 shuffled shoes and priced the bet exactly at every position in each. The average never moves. Undealt cards are a random subset of the shoe, so two cards drawn from them are two random cards of the original shoe, and the expectation at any depth equals the expectation off the top: −2.57%, pinned, all the way down. I checked that by enumerating every subset of every size of a small toy shoe instead of trusting the argument.
What depth changes is the spread. Early on, every composition sits near average and nothing is ever worth betting. Deep in the shoe compositions swing hard in both directions, and the favourable ones are where the whole game is. A player who waits and then bets blind captures 0.00% of the ceiling, measured at every deck count and pay table I ran, which surprises people who assume late shoes run loose.
| cards left | spread | rounds worth betting | best seen |
|---|---|---|---|
| 312 | 0.00% | 0.0% | −2.57% |
| 234 | 0.46% | 0.0% | +0.21% |
| 182 | 1.04% | 1.9% | +3.69% |
| 156 | 1.44% | 5.3% | +6.90% |
| 104 | 2.80% | 17.2% | +16.97% |
| 78 | 4.15% | 25.0% | +22.82% |
5Why no count works, and what does
Every effect of removal is exactly zero, so the answer had to come from a second-order quantity.
Effects of removal are where a counting system comes from: take one card out, see how the bet moves, and the pattern of those movements becomes the tags. Here every one of them is zero, checked in exact fractions at two decks, six decks and eight. Zero down to the last digit, with no cancellation hiding in how the tags are centred. A balanced count also has exactly zero covariance with the bet's value at every depth, so there is no linear signal to find, only ways of failing to find it.
The reason is short. Depth invariance says the average value after removing a random card equals the value before it went. Symmetry says all fifty-two removals move it by the same amount. Together those force the shared amount to be zero.
That explains a detail in the published record. Jacobson's card triggers at a true count of +10 or −10, symmetric in the count, because a wildly lopsided count is weak evidence that the shoe has gone lumpy somehow. It bets 1.27% of hands at +2.28% for 0.029 units per hundred, which is 20.6% of the ceiling. A plain hi-lo trigger at the same threshold does worse, taking 4.4%.
What the bet answers to is the chance two random survivors collide in rank, which depends on the squares of the rank counts. A count holds levels, and squares are beyond what it can carry. Written across thirteen ranks it looks like something nobody could carry, and then it collapses: the whole quantity equals a running tally of pairs already seen, plus the number of cards dealt. Add, for each card that appears, how many of its rank have gone by, so the second seven adds one and the third adds two. That tally and the depth give the value with nothing left over, and betting on it takes 90.2%.
Tracking fewer ranks fails, for a structural reason rather than a statistical one. The number of cards left is pinned by depth, so a handful of ranks looking heavy only means the ranks you aren't watching are correspondingly light, and the two effects on a collision very nearly cancel. Eight tracked ranks captured 1.8%, four and six captured less.
| what you keep | rounds bet | edge when bet | capture | net per 100 rounds |
|---|---|---|---|---|
| exact composition, all 52 counts | 6.53% | +2.16% | 100% | +$4.47 |
| 13 rank counts, pair tally | 6.22% | +2.04% | 90.2% | +$3.10 |
| 13 rank counts, highest minus lowest | 6.07% | +1.38% | 59.6% | −$1.21 |
| Jacobson's published card | 1.27% | +2.28% | 20.6% | −$6.70 |
| hi-lo, |true count| ≥ 10 | 0.19% | +3.30% | 4.4% | −$8.97 |
| 8 ranks tracked | 0.44% | +0.59% | 1.8% | −$9.34 |
The identity, written out
With m for the cards of a rank still in the shoe, D decks, C cards dealt and P the pairs seen so far, the collision quantity is exactly 208D² − 52D − 8DC + 2C + 2P. Every term except P is fixed by the depth, which is why one running tally is enough.
On a flat ladder, where suited and unsuited pay the same, that quantity is the bet's value exactly, with no approximation in it. On 20-10 the suited rung adds a small piece the tally cannot see, and that piece is the whole of the 9.8% the tally gives up.
♠How to play it
One card, and it is the hard one. You keep a single running tally, fed by thirteen small counts of what each rank is up to, and bet when the tally clears the number for your depth. There is an easier published card for this bet, Jacobson's, and it is not printed here because at a $100 maximum it loses $6.70 per hundred rounds against the cost of the seat, needing a $331 maximum before it pays for the chair it sits in. The bet only clears in the last two decks of a six-deck shoe, so for most of the shoe the threshold table is telling you not to bother.
| when | action |
|---|---|
| shuffle | tally to zero, all thirteen rank counts to zero |
| every card you see | add to the tally the number of that rank already seen, then add one to that rank's count |
| so | the second five adds 1, the third five adds 2, the fourth adds 3 |
| bet when | the tally reaches the threshold below for the decks dealt |
| otherwise | no side bet |
| decks dealt | typical tally | bet at | how far above typical |
|---|---|---|---|
| 3.0 | 894 | 918 | +24 |
| 3.5 | 1,218 | 1,234 | +16 |
| 4.0 | 1,592 | 1,603 | +10 |
| 4.5 | 2,016 | 2,022 | +6 |
| 5.0 | 2,490 | 2,493 | +3 |
| 5.5 | 3,014 | 3,015 | +1 |
6Scored on shoes it never saw
Thresholds fitted on half the shoes, banked on the other half: 90.2% of the ceiling.
Every figure on this page is out of sample. I split the 4,000 shoes in half, chose the betting thresholds on one half, and scored them on the other, measuring the ceiling on that same held-out half so a capture is a ratio of two numbers from one sample. Fitting and scoring together would have flattered the thin signals most, which is exactly where a mistake would hide.
The tally holds at 90.2% at three quarters penetration and 91.4% deeper. On the flat 12-12 ladder it reaches 98.8%, because there the suited rung it cannot see is worth nothing.
7The money
$16.30 per hundred rounds one deck deeper than the site standard, where the same card earns $3.10. The cut moves this bet more than the pay table does.
To bet the side you need a seat, and a seat costs a main wager every round while the pair bet is worth making on 8.7% of them. At a $15 table that toll runs $9.60 per hundred rounds and does not shrink when the side unit does. The side maximum has to reach $37 before the play covers its own seat at this cut, and $76 at the standard one. Jacobson's published card needs $331 and never gets there.
At the site's standard 0.75 cut the toll swallows almost everything: $3.10 per hundred rounds on a $426,500 bankroll, and an N0 near 9,200,000 rounds. The scenario therefore runs one deck deeper, and the size of that gap is the finding. I have no data on how often either cut gets dealt on tables spreading this bet, so the range below is a sensitivity and not a claim about what is out there.
The scenario charges the pair bet the entire seat toll, which is the right way to price it for someone sitting down to play this and nothing else. It also means the card needs no blackjack count at all: the main bet stays flat at the minimum, played on plain basic strategy, and the thirteen rank counts are the only thing being carried. A seat already earning its keep another way pays no toll, which is the third row below, but that seat has to be earning it somehow.
the scenario
| penetration | rounds bet | net per 100 rounds | bankroll | N0 |
|---|---|---|---|---|
| 0.75 (the site standard) | 6.2% | +$3.10 | $426,500 | ≈ 9,183,000 rounds |
| 0.8333 (the scenario) | 8.7% | +$16.30 | $113,400 | ≈ 464,000 rounds |
| 0.8333, seat paid elsewhere | 8.7% | +$25.89 | $69,600 | ≈ 180,000 rounds |
Depth, and playing without a seat
Capture holds across the cut: 90.2% of the ceiling at 0.75 and 91.3% at 0.8333, both scored on held-out shoes.
Sitting out and entering only on a trigger removes the toll without needing a second reason to be at the table. It needs mid-shoe entry, and it puts a player in the seat only on the rounds they like.
8The call
A real edge and a real card, priced like work and gated on a deep cut.
Calling this bet uncountable would be wrong. Favourable rounds are common enough to matter deep in a shoe, the exact ceiling is measurable, and a card exists that takes 91.3% of it. The published verdict of near-zero vulnerability comes from asking a linear question about a quadratic bet.
What it asks in return is thirteen exact rank counts held for hours and a cut of at least one deck off six. Miss either and there is nothing here: at three quarters penetration the same card earns $3.10 per hundred rounds, and eight tracked ranks instead of thirteen earn nothing at all.
§The 12-12 table, where suits stop mattering
Pay every pair the same and the tally becomes the whole answer: 98.8% captured, and still not worth playing.
Six decks paying 12 for any pair, suited or not, is a published ladder at a 3.86% house edge. Flattening the top rung changes what a player needs to know. With the suited win paying no premium, two shoes holding the same ranks price identically however the suits fall, and the tally stops approximating the bet's value and becomes it.
Capture goes to 98.8%. The money goes the other way, because a higher house edge means favourable rounds arrive less than half as often: 2.7% of rounds against 6.2%, worth $6.74 per hundred rounds on a $100 bet, which the $9.60 seat toll then swallows. The cleanest table to play is a losing one.
The pattern shows up elsewhere on this site. Pot of Gold's 300-for-1 and 1000-for-1 rungs came out worth −0.51% and −0.21% while carrying most of the variance, and deleting them left the counter strictly better off. Jackpot rungs are where a counter's edge goes to die, which is the reverse of the impression they are printed to give.
| pay table | house edge | rounds bet | edge when bet | capture | net per 100 rounds |
|---|---|---|---|---|---|
| 20-10 | 2.57% | 6.22% | +2.04% | 90.2% | +$3.10 |
| 12-12 | 3.86% | 2.72% | +2.48% | 98.8% | −$2.85 |
§What this verdict depends on
- Six decks with the 20-10 pay table. Wizard of Odds lists it among the published ladders but ties it to no particular property, so whether it is dealt anywhere reachable is unverified.
- A $100 side maximum. Many properties cap this bet at $25, where the seat toll swallows the whole edge at any penetration.
- Penetration is the kill condition. The scenario runs one deck deeper than the site's 0.75 standard, where the same card earns $3.10 per hundred rounds against a $426,500 bankroll. I have no data on what cuts are dealt on tables spreading this bet, so which end of that range applies is unknown.
- Thirteen exact rank counts, held accurately for hours. Nothing on this page tests whether a person can do that, and every capture figure assumes it is done perfectly.
- The bet is priced at every position in the shoe. Live rounds begin at composition-dependent points, which is not modelled here.
- The main game is priced as a flat basic-strategy toll. A player counting the main game as well would offset it, though carrying that count alongside thirteen rank counts is not a serious proposal.
- Bankroll treats the side bet and the main hand as independent.
Last updated 2026-08-10.