Published deviation tables read like commandments: take insurance at a true count of plus 3, stand on 16 against a 10 at zero or higher, double 10 against a 10 at plus 4. Our articles on index plays and the Illustrious 18 cover what these numbers are and how to use them. This article answers the question that usually comes next: where do the numbers come from?
The answer is one idea. An index number is the true count at which two plays have equal expected value. Picture two curves, one per play, each showing expected value against the true count; as the count rises, the curves cross. Below the crossing, one play is better. Above it, the other is. The index number is the crossing point, usually stated as a whole number. A deviation rule in a published table marks a crossing point of two such curves.
Most crossings cannot be computed in your head, because the plays involve drawing rules, dealer bust chances and the exact composition of the shoe. But the break-even point behind one of them can be worked out at the kitchen table, and it belongs to the most valuable single deviation on Schlesinger's published list.
The insurance crossover, derived exactly
Insurance pays 2 to 1 on a side bet that the dealer's hole card is ten valued. Take a one unit insurance bet. If the hole card is a ten, the bet wins 2 units. Otherwise it loses 1 unit. Let p be the share of ten valued cards among the unseen cards. The expected value of the bet is 2p minus (1 minus p), which simplifies to 3p minus 1. Set that to zero and the crossover sits at exactly p = 1/3.
Now measure the starting point. A full six deck shoe holds 96 ten valued cards (the tens, jacks, queens and kings of six decks) among 312 cards, which is 30.8 percent, below one third. That is why basic strategy always declines insurance: from a fresh shoe the bet sits below its crossover, as our piece on why basic strategy never takes insurance explains.
As small cards leave the shoe, the ten share among the unseen cards rises, and somewhere it passes one third. The Hi-Lo insurance index is about plus 3 (per the commonly published indices): the true count at which, on average, the ten share has risen far enough for the bet to break even. The phrase on average is doing real work in that sentence. Hi-Lo tags aces and the small cards as well as tens, so the true count tracks the ten share only approximately, and two shoes at the same count can hold different ten shares. That approximation is one reason indices come from computation rather than a one-line formula, and it is why this article does not attach a ten share to any specific count.
How the rest of the numbers are computed
Every other index is found the same way in principle and a completely different way in practice. For a hand like 16 against a 10, you would need the full expected value of hitting (which can mean drawing several cards) and of standing, at every true count, under the exact rules in force. That is typically done by computer: either simulation, playing enormous numbers of hands with both plays forced at each true count, or combinatorial analysis, which enumerates the possible card combinations directly. In either case the two plays are compared at each true count and the crossover is recorded. Published tables usually give that crossover as a whole number, so a player can compare it with a true count computed at the table.
This is worth stating plainly because it sets the right expectations: the numbers are computed estimates of a crossing point, not exact physical constants. Small differences between careful computations move a crossing most when it happens at a shallow angle, where the two plays are close over a wide range of counts.
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Why published lists disagree
Once you know that an index is a crossing point under specific conditions, the disagreements between published tables stop being mysterious. Four inputs move the numbers.
- Deck count. The same play can cross at a different count in a single deck game than in a six deck shoe, partly because the removal of a few cards shifts a single deck composition more.
- House rules. The soft 17 rule and other rules can shift the curves. The Fab 4 surrender indices, for instance, differ between games where the dealer stands on soft 17 and games where the dealer hits it, per the commonly published lists.
- The counting system. A Hi-Lo true count and a Zen true count are different scales. An index computed for one system does not transfer to another; using a Hi-Lo index while running a KO count is a category error, not an approximation.
- The rounding convention. The true count you actually use is a whole number produced from a fraction, and whether you floor, truncate or round that fraction shifts every trigger by up to a point. Our piece on rounding the true count works a single hand through all three conventions.
So when two sources give different numbers for the same play, the first question is whether they are answering the same question. Often they are not: different decks, different rules, different systems, different rounding.
The crossover that sits at zero
Standing on 16 against a 10, at a true count of 0 or higher, is the most used hand deviation on the Illustrious 18, and its index of zero tells you something. The crossing sits almost exactly at the neutral shoe: at a fresh, average mix of cards, hitting and standing on 16 against a 10 are close enough that the chart's hit is a thin decision, and the balance tips to standing with even a modest surplus of tens. Our deep dive on 16 against a 10 describes the same closeness from the basic strategy side: hitting loses slightly less money than standing, and the margin is small. Hands like this, where the basic play wins by a hair, are exactly the hands that make good deviations, because the crossing is close enough to reach in ordinary play.
Why learners start with a ranked list
Complete index tables run to well over a hundred entries, and memorizing them all is a poor use of study time. Don Schlesinger, in his book Blackjack Attack, ranked the deviations by how much each is worth and published the shortlist that became known as the Illustrious 18: the eighteen deviations that deliver most of the available playing gain. A companion set, the Fab 4, covers the most valuable late surrender deviations. The ranking is why a learner starts with those lists rather than the full tables, and our guide to memorizing index numbers builds a study order from it. The value ordering also explains why the insurance index, whose break-even point is derived above, sits first on the list: the situation recurs every time an ace shows.
Learning the crossovers
In the 21 Trainer app, the Pro upgrade's index deviation training covers the Illustrious 18, the Fab 4 and extended indices, with an index quiz, alongside the upgrade's four counting systems: Hi-Lo, KO, Omega II and Zen.
The standing caveats that accompany any counting article apply to deviations too. Counting with your head, index numbers included, is legal in the United States. Casinos are private businesses that may refuse service to players they suspect of counting. And no technique, index tables included, removes the risk from the game. What the numbers give you is the knowledge of when the better play switches, which is a piece of applied probability worth learning for its own sake.
Train the deviationsIndex deviation training covering the Illustrious 18, Fab 4 and extended indices, in the Pro upgrade.
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Frequently asked questions
What is an index number in card counting?
The true count at which two candidate plays have equal expected value. Below that count one play is better; at or above it the other play is. For example, the Hi-Lo insurance index is about plus 3, the point where the insurance side bet breaks even on average. An index is a threshold, not a strength: it says when to switch plays, not how much the switch is worth.
How are index numbers calculated?
Typically by computer, using either simulation or combinatorial analysis over very large numbers of hands. The two candidate plays are compared at each true count, and the index is where the better one switches. Published tables usually give the crossover as a whole number, so it can be compared against a true count you compute at the table. Nobody derives them by hand at the felt.
Why do different sources give different index numbers?
Because the inputs differ. Deck count, the soft 17 rule and other rules move some crossovers; each counting system runs on its own scale, so a Hi-Lo index does not transfer to KO or Zen; and the convention used to turn a fractional true count into a whole number shifts triggers by up to a point. Indices are always specific to a system, a rule set and a rounding convention.
What is the most important index number to learn?
Insurance at a Hi-Lo true count of about plus 3, per Don Schlesinger's published ranking. It ranks first because the situation comes up every time an ace shows. Standing on 16 against a 10 at a true count of 0 or higher is usually listed next; its index sits at zero because hitting and standing are so close in a neutral shoe.
Is it legal to use index numbers in blackjack?
Yes. Card counting done in your head, index numbers included, is legal in the United States: no federal or state statute makes mental card counting an offense, and devices are a separate legal matter. Casinos are private businesses that may refuse service to players they suspect of counting, and no technique removes the risk from the game. That is the honest frame for the whole skill.