Vivian and I decided to spend some of our Christmas holidays in Las Vegas because Las Vegas, to attract gamblers, had turned itself into a bargain vacation spot. As a twenty-six-year-old with a PhD in mathematics, I was earning too little at UCLA to treat money casually. I also believed then, as I do now after more than fifty years as a money manager, that the surest way to get rich is to play only those gambling games or make those investments where I have an edge. Since I knew of no one who had ever found a way to beat the casinos, gambling in Vegas wasn’t on my list of priorities.
Seeing Las Vegas back in 1958, I did not imagine today’s glittering strip of giant high-rise hotels, jammed together helter-skelter, fronted at all hours by multi-lane, gridlocked traffic. Legendary casinos such as the Sands, Flamingo, Dunes, Riviera, and Tropicana are gone, the mob and cash-skimming operations replaced by multibillion-dollar public companies. Back then the long, straight, uncrowded highway had a dozen or so one-story hotel-casino complexes scattered on either side with hundreds of yards of sand and tumbleweeds separating them.
Just before we left for our vacation, my colleague Professor Robert Sorgenfrey told me about a new strategy for playing blackjack that claimed to give players the smallest house edge of any casino game. Next best was baccarat, with a house advantage as low as 1.06 percent, and then craps, with some bets costing just 1.41 percent. The new figure of 0.62 percent for the house edge in blackjack was so close to even that I planned to risk a few dollars for fun. Devised by four mathematicians during their time in the army, the strategy covered several hundred possible decisions a player might face. I condensed its main features onto a card that fit into the palm of my hand. My only experience in a casino had been my earlier adventure putting a few coins in a slot machine.
After settling into our hotel room, we headed for the casino. Weaving past drinkers, smokers, and slot machines, I found two rows of blackjack tables, separated by an aisle or “pit” complete with reserves of chips, extra cards, and cocktail waitresses who offered alcoholic nirvana to the marks, or suckers, all of whom the pit boss monitored closely. It was early afternoon and the few open tables were busy. Managing to get a seat, I plunked down my entire stake—a stack of ten silver dollars—on the green felt table behind my “betting spot.” I didn’t expect to win, since the odds were slightly against me, but as I expected to build a device to successfully predict roulette and had never gambled before, it was time to get casino experience. I knew virtually nothing about casinos, their history, or how they operated. I was like a person who had glanced at recipes but never been in a kitchen.
The game I was about to play, blackjack, or twenty-one, was basically the same as the Spanish game of twenty-one, referred to as early as 1601 in a story by Cervantes. During the mid-eighteenth century, as part of the European gambling craze of that time, the French called it vingt-et-un. Later, when the game was introduced in the twentieth century to US gaming establishments, bonuses were sometimes offered for special combinations of cards, among them a ten-to-one payoff if a player’s first two cards were the Ace of Spades plus one of the two black Jacks, a “blackjack.” Though the bonus was soon rescinded, this name for the game stuck, and any two-card total of 21—that is, any Ace plus any Ten-value card—is now called blackjack as well.
The action starts when the players place their bets on “spots” in front of them, after which the dealer gives two cards to everyone and also two cards to himself. The first of the dealer’s cards is dealt faceup for everyone to see, and the second is placed facedown under it. Then, starting with the player on his left, the dealer asks each participant in turn how he wants to play his hand.
The goal of both player and dealer is to draw cards to get as close to a total of 21 as possible without going over. Anyone who does so “busts” and immediately loses. Aces count as either 1 or 11 at the option of the player. Ten, Jack, Queen, and King all count as 10, the so-called Tens or Ten-value cards. The other cards, 2, 3, and so on through 9, count as their face value. The dealer typically must draw until he has a total of 17, then he must stop or “stand.” A player may stand at any time. The dealer’s edge is that the player must risk busting first, losing immediately, even if the dealer later also goes over 21, though in effect they have tied. Because he loses when they both bust, a player who follows the dealer’s strategy for his hands has a disadvantage of about 6 percent.
On the other hand, the dealer has fixed rules to follow and the player doesn’t, giving the player more choices than the dealer, and this flexibility to choose a different way to play his cards can have a remarkable impact. Among these choices, before drawing or standing, a player with two cards of the same value, such as a pair of 9s (9, 9), may split them into two new hands, adding another bet equal to his original one, and getting a second card on each. He then plays out the two hands one at a time. Not all pairs should be split. Splitting a pair of 8s, for example, usually turns out to be good, but splitting Ten-value cards does not. Alternatively, the player can elect to turn his first two cards faceup, double his bet, and get exactly one more card. This is called doubling down. Unlike the dealer, the player is also free to draw or stand on any total of 21 or less.
When I sat down, the players at my table had been losing heavily and continued to do so. I worried about consulting the tiny strategy card in my palm. Would I be kicked out? Forbidden to use the card? Instead, ridicule was the problem. When I held up the game by consulting my strategy card, the dealer made patronizing “helpful” suggestions on how to play my hand and indicated to the crowd that he was dealing to (and with) someone just in from the farm. Snickering at my unorthodox bets, onlookers wondered who would split a pair of lowly 8s, doubling the amount of money at risk, when the dealer’s upcard was the powerful Ace? What fool would stand on a piteous total of 12 against a dealer’s weak upcard of 4? Surely my ten silver crumbs would soon be swept away by the dealer. Or would they?
Playing with unhurried deliberation, I managed to keep my stack of chips even. Then a strange thing happened. I was dealt an Ace and a 2, and since Aces can be counted as either 1 or 11, my total was either 3 or 13. Next, I drew a 2, and then a 3. I now had Ace, 2, 2, 3, for a total of 8 or 18. The dealer had a 9 up, and might or might not have had a Ten hidden for a total of 19, but 18 was a pretty good hand. Surely only a fool would draw again and risk the destruction of such a good hand. The strategy said to ask for another card. I did. It was with no little satisfaction and several tsk-tsks that the amused onlookers saw me get a 6. Now I had to value the Ace in my hand as a 1, for a total of 14! “Serves him right,” said a bystander. For my sixth card, I drew an Ace, which brought my total to 15. “You deserved to bust,” muttered a kibitzer. Now I drew my seventh card. It was a 6! I now held Ace, 2, 2, 3, 6, Ace, 6 for a seven-card total of 21. This is a rare event.
After a moment of shock, several people in the crowd said I had a $25 bonus coming. The dealer said no—it was paid at only a few places and those were in Reno. Unaware that any casino even had such a rule, I thought it might be amusing to create the impression that I had indeed sacrificed my 18 because I foresaw the seven-card 21. And who knows. They might even pay me. Of course they did not. But the amusement and patronizing attitude of some bystanders changed to attentiveness, respect, and even awe.
Fifteen minutes later, I had lost $8.50 of my $10 stake and quit. But now, to Vivian’s dismay, I was hooked on blackjack—though not in the usual sense. The atmosphere of ignorance and superstition surrounding the blackjack table that day had convinced me that even good players didn’t understand the mathematics underlying the game. I returned home intending to find a way to win.
Had I been more knowledgeable about the history of gambling and the centuries of effort devoted to the mathematical analysis of games, I might not have tackled blackjack. To anyone who observes the glittering Las Vegas Strip and the gambling boom that has brought lotteries and casinos to most US states, clearly a lot of people are losing a lot of money—tens of billions annually. Furthermore, for most casino gambling games, mathematicians had proved no system of varying bets could blunt the casino’s edge. Generations of gamblers had been seeking the impossible. Players are confused about the inevitability of losing in the long term because they each play for a comparatively short time, which allows some of them to be lucky winners.
This is true for any game where you can compute the edge and the payoff does not depend on the results of previous bets or the wagers of other players. Coin tossing, craps, keno, roulette, and the money wheel are examples, provided you don’t have, for instance in the case of wheels, the help of a prediction device. Horse racing and the stock market are different, because you can’t compute the probabilities and because other players’ bets affect the payoffs.
The belief that casinos must come out ahead in the long run was supported by conventional wisdom, which argued that if blackjack could be beaten, the casinos would have to either change the rules or drop the game. Neither had happened. But, confident from my experiments that I could predict roulette, I wasn’t willing to accept these claims about blackjack. I decided to check for myself if the player could systematically win.