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The guidelines for hitting are rather straightforward. If the dealer shows a 2 or 3, you continue to take a hit until you have a hard 13 or a soft 18. If the dealer shows 4, 5, or 6, you continue to take a hit until you reach a hard 12 or a soft 18. If the dealer shows 7 or 8, you continue to take a hit until you have a hard 17 or a soft 18. It the dealer shows anything higher than 8, you continue to take a hit until you have a hard 17 or a soft 19.

Dealer Shows Hit Until You Have
2 or 3 hard 13 or soft 18
4, 5, or 6 hard 12 or soft 18
7 or 8 hard 17 or soft 18
9, 10, or A hard 17 or soft 19

Though there is no mathematical “proof” of these principles there is actually some simple logic to them. Don’t forget that you are also playing the odds based on billions of simulations of blackjack hands. Let’s look at some of the logic.
? If the dealer shows a 7 or above, then the most likely two-card total is 17 or above (with a 10 or an A in the hole), so you are going to have to take a card on any total under 17 or likely lose.
? When the dealer shows a card less than 7, the two-card total will likely be less than 17 (it can be exactly 17 with a 6 and an A), and the dealer will be forced to take another card. Since there are more 10’s in the deck than any other denomination, the dealer will have a fairly high probability of busting and you will win.
? If you were to take a card with a total between 12 and 16 you would be likely to bust. In situations like this the proper play is to let the dealer pull. If the high card shows up and there is a high card in the hole, you will win.
? Hit a total of 12 against a dealer 2 or 3. I’ve seen books that tell you to stand in these situations. They are wrong. You must take a card.
? Hit a 16 against a dealer’s 7. Many inexperienced players have trouble believing that this is the proper play but it is. Countless computer runs have proved it again and again. From the players point of view a total of 16 is no better than a total of 12; you can win with such totals only when the dealer breaks. Besides, there are still five cards that can help out a 16 (A, 2, 3, 4, 5).
? You take a card whenever you have A, 6 (unless you double down) and you hit an A, 7 against a 9, 10, or A. It’s true that you will sometimes find yourself going “backwards” and have a hand that is “weaker” than you just had. However, computer simulations consistently show that this is the proper play.
? It may come as a surprise to inexperienced players, but 18 is not a strong hand when facing a dealer 9, 10, or A.

Since the essential features of basic strategy were developed, a number of refinements have given us the current optimal set of principles for standing, hitting, doubling, splitting, and surrendering. These, along with the other more sophisticated forms of play were worked out by using Monte Carlo techniques based on the analysis of literally billions of hands. If I tell you that you should hit a total of 16 against a dealer’s 7, there is no specific mathematical proof behind this recommendation. It emerged from an analysis of the several million times this situation emerged in the Monte Carlo analysis of the game. Hitting a 16 against a 7 loses less often than standing. Sure, following this advice produces a bust on a lot of these hands, but analysis shows, utterly compellingly, that if you don’t hit his hand you are more likely to get beat by a higher total — like 17.
When possible, I will give a logical analysis of particular aspects of basic strategy, but there are going to be situations where the reader is just going to have to accept the outcomes of the Monte Carlo analysis. The following description of basic strategy is based on the multi-deck game found in several Atlantic City and Las Vegas casinos, where the dealer stands on a soft seventeen, pairs may be re-split once, doubling down is permitted after a split, and the player may double down on any two cards. Other games require some minor adjustments that I’ll note where appropriate. However, you should never give up an edge against the casino. I highly recommend playing only where the rules are more favorable to the player.