BERNS ON HEARTS Copyright 2002 Gerald M. Berns 1995-2002 Introduction The game of Hearts (also, and more descriptively, known as Black Lady) is so simple! The dealer deals the deck one card at a time to four players. From the 13-card hand, on three of every four deals, each player picks three cards to pass to an opponent. The first pass is to the player on the left; on successive deals the pass is to the right, then across, and then no pass (a "Hold" hand). Then the passing cycle repeats. Play begins with the player at the dealer's left leading a card (a common variation: the player who has 2 of clubs (2C) leads it). The card led establishes the trick's suit, and each player in turn plays a card in that suit, if she has one; if not, a player may sluff any card. The player whose card is highest in the suit led wins the trick and leads the next trick. However, a heart may not be led until hearts are "broken" someone sluffs a heart or plays the queen of spades (QS) or the trick-leader's hand contains only hearts. Hearts is a negative scoring game: points captured are (usually) bad, not good. Capturing each heart counts one point against the capturer, and capturing QS counts 13 points against. However, there is a hitch. If a player captures all 26 points, she "Shoots the Moon"! For this achievement she receives no points and her three opponents each receive 26 points; however, if adding 26 points to each of her opponents scores does not put her in the lead, the Shooter subtracts 26 points from her score. The game ends when a player's score reaches or exceeds 100, and the player with the lowest score wins (ties are allowed). These are the rules and variations that Killer Hearts supports (some players use other variations). Hearts is so simple to understand and play that it may seem almost childlike. Consider that there is no bidding, no trump, no wild card, no melding, no collecting, no partner with whom to communicate and coordinate, no doubt about whether a particular card has been dealt, and no arcane scoring system. Unlike Bridge, it puts no premium on memorizing conventions and systems. Unlike Setback, it puts little premium on courage and daring. Unlike Checkers and Chess, it puts no premium on depth of vision down a move tree. Moreover, in three of every four hands a player can pass away her three most vexing cards. Despite its simple rules, Hearts is a surprisingly complex game. To appreciate its complexity, first consider the goals of playing a hand in several other card games: Bridge: (Bid and) win the maximum number of tricks Setback: (Bid and) win the maximum number of points Casino: Win the maximum number of points Canasta: Make the maximum number of melds The goals of these games are straightforward and invariant. Now consider the goals of playing a hand of Hearts, most important first. If currently low-scorer: Maintain winning position and Play so that an opponent goes out Expand scoring margin over second-place player Play so that high-scorer takes maximum number of points Play so that second-low player takes maximum number of points If not currently low-scorer: Become low-scorer Protect opponent with very high score from going out Prevent Moon Shoot that wins game Prevent Moon Shoot that produces new low-scorer Play so that low-scorer takes maximum number of points Win the minimum number of points Some may object to the presence, absence, or order of various goals. Others may object that some goals should be further caveated, that they do not apply when a player's score is very high. But that there are multiple, score-sensitive goals is undeniable. Goals run the gamut from straightforward to subtle. They can vary from hand to hand and even within a single Hearts hand! With a multiplicity of shifting goals comes complexity. Hearts has surprisingly subtle strategies in keeping with its many goals. They give the game a captivating charm. Then, too, there is the "catastrophe" (as in Catastrophe Theory) at the center of the game. If you succeed too well in ducking points, an opponent may collect all of them (Shoot the Moon), which may cost you 26 points. If you attempt to Shoot the Moon and almost succeed, your score can increase by as much as 25 points instead of decreasing by 26. That is, if you are too good at avoiding points or not good enough at Shooting the Moon, look out! The game is called Hearts, but it is really all about handling QS. In this paper I explore some of the nuances and strategies that make playing Hearts so challenging and so much fun. The Pass The three-card pass enables you to improve the hand that random chance dealt you. The question is: Which are the three best cards to pass? (This is one of the most important questions that the game asks, because studies with Killer Hearts show that, in general, a larger change in performance results from changing passing strategy than from changing playing strategy.) High cards take tricks, taking tricks captures points, and points count against you. So your first thought might be to pass your highest cards. But wait! Passing your three highest cards to a skilled opponent may enable her to Shoot the Moon! Conversely, passing three low cards may enable an opponent to avoid taking tricks and cause you to take tricks and points. What to do? Many players (and Hearts-playing computer programs) are so concerned about an opponent Shooting the Moon that they usually pass at least one beatable heart. The idea is that the recipient cannot Shoot the Moon with a beatable heart in her hand, because the passer or someone else will take the trick when she plays it. This thinking rests on three premises: * Skilled players frequently Shoot the Moon * The pass receiver does not have many heart winners (if she does, the passed heart will not hurt her and may actually help: it is one less heart that the Shooter needs to pull, and it may help her avoid taking tricks if she decides not to Shoot) * Someone else will take a trick on which a beatable heart is played But attempting to Shoot the Moon presents a sizable risk, because a close miss is very expensive. So, even skilled players approach it with caution. Capturing all the points except the last heart results in a 51 point swing against the would-be Shooter: 25 points against versus 26 points for. Furthermore, even though a heart trick only counts a few points against the taker, there is no guarantee that someone will take a heart-bearing trick. Experienced players often duck such tricks (for various reasons, including "Don't worry, someone else will take it!"). To guarantee that the recipient cannot Shoot the Moon, there is only one way to pass: pass a heart that you can beat and retain AH, KxH (KH and a lower heart), QxxH, JxxxH, etc. Play your stopper when the heart you passed is high on a heart trick and the points have not yet been split. Of the three premises, only the second is consistently encountered when playing with people. Shooting the Moon is a relatively infrequent event: against expert competition, even the best players cannot do it more often than about once every 32 hands, on average. So why habitually use your pass to protect yourself against the pass recipient's Shooting the Moon, especially when that protection is weak? You can purge your hand of only three cards. Why spend a third of your pass in mechanically passing a heart? A policy of passing a heart means that often another problematic card must stay in your hand. Selecting the three best cards to pass (which may also be thought of as the three worst cards to keep) is an intelligent process that must consider many factors, including scores and the passer's degree of risk aversion. Several factors are mutually contradictory, so it is a subjective process and experts do not always agree on the result. Nevertheless, there are some basics to consider. The first concerns the all-important spade suit. Suppose you are dealt QS (a.k.a. "the Lady," as well as many not-so-polite names). The numero uno question is: Should you keep it or pass it? If you pass it, your opponents cannot repeatedly lead spades and force you to play it and possibly take it (because only AKS have more capturing strength). But passing QS means giving up control over who captures her. Furthermore, passing QS may cause the recipient to view your deed as a social act on a par with spitting on her sandwich. She may react harshly, looking to return the unwelcome gift to you post haste. Keeping QS can affect your pass. Suppose you also hold AQ94C. If you do not keep QS, you might pass AQC leaving the 94C doubleton. But, keeping QS, you might instead pass 94C leaving the AQC doubleton. When play begins, you take the first club trick with the AC and lead back the QC. Now, possibly as early as trick 3, you have a club void through which you may be able to sluff your QS! Furthermore, if QC wins, you retain the lead and can continue to purge your winners. In so doing you may even gather a heart, killing your opponents' Shoot hopes. Holding QS in your hand gives you immunity from unexpectedly winning her in a trick. Controlling QS means more than simply sluffing it at the first opportunity. Assuming that you are playing Hearts to win and not just for the fun of participating, and that you have enough spades to protect your QS, controlling QS means planning to give it to a particular player. Who your target should be is entirely score-dependent. If you are not the low scorer (i.e., not currently winning the game) and the game is well along, you target that player, trying to make her the ex-low scorer. Although giving QS to any player whose score is lower than yours may appeal, the hard fact is that you cannot win until the low scorer is dislodged. On the other hand, if you are not winning the game you must not give QS to a player who is at the brink of elimination, for eliminating her eliminates you! If you are the low scorer, you have more potential targets: the high scorer to push her score to 100, a close competitor to increase your winning margin, and, in a crunch, anyone but you. It nets out like this: You should always keep and thereby control QS if you can tolerate the risk. When is the benefit of holding QS worth the risk? There is more to this question than merely assessing how many spades your opponents hold and their eagerness to pull your QS out with them. Remember that one may sluff any card when void in the suit led. Therefore, the risk analysis reduces to a single compound question: Can your opponents lead spades often enough to pull QS out of your hand before someone leads a suit in which you are void in a trick that one of your targets will probably take? Now we are tickling the essence of the pass. You can tolerate holding QS with fewer protective spades if you have a void, and the earlier you have the void the better. The concept of a suit void is larger than a discussion of passing QS. If you pass to make a void, you are in a position (unless the incoming pass fills the void) to sluff any card whenever that suit is led. This is an excellent defensive position. If you think someone (not just your pass recipient) is trying to Shoot the Moon, you can sluff a heart on another player's trick. Or you can use the void to sluff problem cards or to create additional voids. And, of course, you can sluff QS. Defensive Hearts, wherein you ensure that points are split among at least two players, often depends on the management of voids. # Suit Distribution Frequency (%) 1 13 0.00 2 12 1 0.00 3 11 2 0.00 4 11 1 1 0.00 5 10 3 0.00 6 10 2 1 0.00 7 10 1 1 1 0.00 8 9 4 0.00 9 9 3 1 0.01 10 9 2 2 0.00 11 9 2 1 1 0.01 12 8 5 0.00 13 8 4 1 0.04 14 8 3 2 0.10 15 8 3 1 1 0.11 16 8 2 2 1 0.19 17 7 6 0.00 18 7 5 1 0.10 19 7 4 2 0.36 20 7 4 1 1 0.39 21 7 3 3 0.26 22 7 3 2 1 1.88 23 7 2 2 2 0.51 24 6 6 1 0.07 25 6 5 2 0.65 26 6 5 1 1 0.70 27 6 4 3 1.32 28 6 4 2 1 4.70 29 6 3 3 1 3.44 30 6 3 2 2 5.64 31 5 5 3 0.89 32 5 5 2 1 3.17 33 5 4 4 1.24 34 5 4 3 1 12.93 35 5 4 2 2 10.57 36 5 3 3 2 15.51 37 4 4 4 1 2.99 38 4 4 3 2 21.55 39 4 3 3 3 10.53 Table 1. The 13-Card Hand's 39 Suit Distributions and Frequency of Occurrence. Only 39 suit distributions are possible in a 13-card hand. Table 1 shows them, along with their frequency of occurrence (calculations courtesy of Brian M. Berns, my son). Suit distribution #1 in Table 1 is all cards in a single suit, which occurs in less than 0.01% of hands, and #39 is the "normal" distribution of one four-card suit and three three-card suits, which occurs in 10.53% of all hands. Note, however, that four distributions are more frequent, and that just five distributions (#34, 35, 36, 38, 39) account for nearly 75% (71.09%) of the hands. Table 1 holds for hold hands. Hold hands occur only 25% of the time, so they tend to get overlooked. That is unfortunate. Table 1's distributions and frequencies of occurrence can be used to answer an often-asked question about hold hands. If a player is not dealt QS, how many times can she lead the same non-spade suit without serious risk that the person holding QS is void in the suit (the implication being that, if the QS-owner is void in the suit, she will unload her cargo)? Table 1 shows that hands with at least one void comprise 5.04% of all hands, and that hands with a void or singleton comprise 35.55% of all hands. Recognizing that the short suit in question is spades 25% of the time and that a player is dealt QS 25% of the time, we find that the player dealt QS is also dealt a void 0.95% of the time, and is dealt a void or singleton that is not spades 6.67% of the time. So, in a hold hand a player can lead the same non-spade suit twice with less than 7% risk that the QS-owner is void in the suit. Further reducing the risk are the considerations that someone other than the trick-leader will win the trick, and that the QS-owner may not be interested in putting QS on the trick-winner. Experience with Killer Hearts supports this conclusion. Table 1 shows, somewhat surprisingly, that a three-card pass to make at least one void can be made from every hand! A pass to leave two voids can be made in 29 of the 39 possible hands (which collectively occur in 14.95% of all hands, or about once in every seven hands). This is not to mention the singletons and doubletons that passing can produce. The three-card pass can wreak havoc with suit distributions! The trick in Hearts is to take maximum advantage of skewed distributions in general and voids in particular. The only spades frequently passed are AKQ. AKS are passed because they can capture QS. Players dealt one or both of AKS have a similar keep-or-pass calculus as with QS, and they often pass one or both. So, if you pass QS, you may receive AS or KS in the mail! However, if you keep QS, receiving AS or KS adds to your spade holding and is a significant asset, not a burden. There is another consideration when you are dealt QS in a short spade suit, if you are playing against experts. If you are not the low scorer, and if you are passing to someone who is also not the low scorer, she may be in a better position to pin QS onto the low scorer (e.g., she may have a longer spade suit or a non-spade void, or she may sit last on a trick when the low scorer leads). In this case, and only in this case, you should consider passing her QS and at least one other spade to mitigate her risk and soften the pass. But if she is the low scorer, or you are not sure of her skill and experience, or you are passing to a computer player (other than Killer Hearts' Better or Best Player), I do not recommend such a pass. So, to pass QS or not to pass? I suggest the following guidelines for defensive hands. Pass QS if the hand dealt to you contains: * Two or fewer other spades * Three other spades, the pass is to the right, and the player to your right is not low-scorer; in this case you always play immediately after the QS-owner, which means that you can safely play your big spades when spades are led Otherwise, keep QS. Remember that these are just guidelines. Bear in mind that the pass coming your way may fill your voids! If you expect your target to be so elusive that you may have to hold QS for many tricks, you may need more spades. Make a special effort not to pass QS to the low scorer, for she is the last person you want controlling QS. Most people who pass a heart pick a beatable one, as we have discussed. However, do not be afraid to pass your highest heart. The probability is low that the pass-recipient has enough other hearts without losers to mount a serious Shoot attempt. Note, however, that passing AH by itself does not devalue the recipient's heart void if she plans to Shoot. That a heart cannot be led at will makes a heart void less effective defensively than all other voids. Because a heart is usually not led until after all other suits have been played at least once, you should not expect a heart void to be useful until midway through a hand. On the other hand, a spade void is even more valuable than a diamond or club void, because spades are usually led early and often. Of course, passing to achieve a spade void is most risky, as the pass coming your way may contain some of AKQS! Only when you are dealt a spade singleton, or a doubleton or tripleton from among AKQS, should you consider passing all your spades. My rule of thumb for all modes of play is that a void and a doubleton are better than two singletons, a void and a tripleton are better than a singleton and a doubleton, and a singleton and a tripleton are better than two doubletons. However, I do not recommend passing a deuce or a trey, unless your need is urgent (e.g., you have decided to keep QS with few spades). Table 2 lists some personal passing preferences for defensive hands (as opposed to Shooting hands) that do not have QS. Before Pass Preferred Result Not Preferred (to pass 2) T86 KQ53 6 KQ53 T86 53 (to pass 2) AJ QJ4 void QJ4 AJ 4 (to pass 2) 862 Q8 862 void 2 Q8 (to pass 2) 72 AT5 2 T5 void AT5; 72 5 (to pass 2) A32 KJ53 32 J53 A32 53; 2 KJ53 (to pass 1) KJ7H J7H K7H Table 2. Personal Passing Preferences for Defensive Hands without QS. Until now we have considered the pass only as a defensive operation. This is appropriate because Hearts is essentially a defensive game in which players normally try to avoid taking points. Shooting the Moon, however, is an offensive operation. It is the only playing mode in which one sets out to get more than one point. Even seasoned players approach Shooting the Moon with trepidation, because of the associated risk. Inexperienced players tend to believe that the only way to Shoot the Moon while keeping the risk tolerable is with a hand having a long, strong run of hearts. This looks like the easiest and surest way to capture all the hearts, and, should they fail to get QS, their flirtation with Shooting only costs 13 points. But a solid high heart run occurs infrequently. Experienced players do not turn away from such a heart run, but they do not wait for one, either. A subtler approach to Shooting the Moon is, perversely, to avoid holding hearts entirely. With a hand that contains one or more long runs of high cards (and, preferably, also low cards, so that you can avoid taking tricks if the Shoot goes awry), consider voiding yourself in hearts if you are dealt three or fewer heart losers. Such a hand occurs much more often than a long, strong heart suit. If a pass fills your heart void, then you have only those hearts to handle. If you play a heart and lose it, then cancel your Shoot. On the other hand, your Moon Shoot may be launched if it wins. The word to remember when deciding your best passing strategy is: Voids! Playing the Hand As mentioned earlier, the name of the game may be Hearts, but the game is all about QS! To a very large and perhaps surprising degree, hearts themselves play a small role. That each heart captured costs one point is interesting but usually unimportant. Hearts games revolve around QS, because its value is 13 points. In the large majority of Hearts games, the winner is the player who takes QS least often, and the last-place finisher is the player who takes her most often. If the rules changed so that a heart had no point value, Hearts would be played almost the same way. Therefore, to win at Hearts is simple: Avoid taking QS! Unfortunately, trying to avoid QS is what everyone else is doing, too. What you must expect and accept is that, no matter how well you play, the Black Lady usually embraces every player more than once in every game. No one avoids her for too long. Winners at Hearts just avoid her more often than others. Hearts may be a simple game, but it is not simple-minded. The way I (and, therefore, Killer Hearts) view the game, it has eight discrete modes of play, some not so obvious: Playing as low-scorer Playing to lie low Playing to "get" the low-scorer Playing to Shoot the Moon Playing while equivocating about Shooting the Moon Playing to stop a Moon Shoot Playing to help a high-scorer Shoot the Moon Playing to protect a high-scorer These modes are primarily dependent on your score relative to the scores of your opponents. Other Hearts-playing programs that I have examined do not seem to consider score-dependency an important aspect of the game. Playing a Hearts hand consists of answering one of three questions on every trick: What shall I lead? Shall I follow high or low? What shall I sluff? Each playing mode may require a different answer. To excel at Hearts, you must be familiar with all modes. Playing Defensively: Who Has the Queen? To win at Hearts, one must avoid taking QS as much as possible. This becomes a much easier task if you know where she sits. However, in most cases when you do not hold QS yourself and have not passed her, determining who holds QS requires slicing through considerable ambiguity. The ability to infer where QS sits is a major factor in separating excellent Hearts players from cannon-fodder. If you have figured out who has QS, your problem is eased. You can play as high as you like after she has played. But, if you win the trick, the problem of selecting a safe lead remains. The best situation is to have QS yourself and let your opponents worry about who has her. And you know where she is if you passed her. If you are passed KS or AS, the passer did not have QS before she received her pass, because a person holding QS keeps all her spades as protection. The player sluffing on a trick that the low-scorer is winning (or is likely to win) does not have QS (or is trying to Shoot the Moon). A player leading a sequence of unusually high cards probably has QS, or is Shooting and wants her. A player following with unusually high cards may have QS or believe that the person with QS has already played (a case in which you learn by observing an opponent's play). A player leading the AS or KS has her, is Shooting, is bluffing a Shoot, or has lost her mind completely. Playing Defensively: General Considerations New players are acutely aware of this logical progression: High cards win tricks Winning tricks catch QS A common neophyte strategy is: never take a trick! This can work well early in a hand, but by the end of the hand the newcomer's low cards have all been played, and guess what happens? You do not want to take more tricks than necessary, but every player usually takes tricks in every hand. However, the better players take tricks early in the hand, hoping that the holder of QS is not yet void. The best situation in which to take a trick when it is late in a hand is when you play last and have an "out-card" (one that will lose a trick) available to lead. If you do not know who has QS, the earlier you play on a non-spade trick, the greater the risk of playing high, and leading high is the riskiest. The earlier you play on a trick, the more players behind you who might have a void, QS, and the inclination to drop her on you. If a suit in which you have five or more cards is led for the first time, do not play high unless you sit last. With the possibility of sluffing it through a void, you may never have to lead or follow with the highest card in this suit. On the other hand, if you have four or fewer cards in the suit and poor chances to develop an early void elsewhere to sluff winners in this suit, you may have to swallow hard and go high on a trick. Much better to go high early in a hand than late. If you do not hold Qs or know where she sits, selecting a card to lead requires care, because you cannot know in which suits the player holding QS is void. Dastardly players have even been known to drop QS on the first trick (rules permitting)! There is no foolproof defense against this except a] mindlessly leading spades, b] leading the lowest unplayed card in a live suit (one in which at least one opponent is not void), or 3] standing in good with the Lord. Hold hands have more reliable distributions. A hold hand with a void occurs in just 5.18% of the cases (see Table 1). So, an opponent is dealt a Hold hand with QS and a diamond or club void with a frequency of just 0.86%, which is once per 116 hands. Here are guidelines for leading when you do not hold QS and it has not yet been played: * Lead spades if you know it is safe (i.e., it will not harm the wrong player) * Never lead a high card in a long suit if it is long in your hand, it is short in others If you do not know who holds QS: * Never lead high in the suit in which you were passed three cards; your passer may now be void and have a reward for your obliviousness * Never lead the suit in which the player on your left passed you two or more cards, for she may now be void and have a similar reminder The riskiest holding in the game of Hearts is a short spade suit containing AS, KS, or QS. Next risky is a non-spade suit of four or more cards in which the lowest is a 6 or higher and you do not hold QS. There are four cards lower than a 6, and you should prepare to see them at the most inopportune time! With such a holding you need to be lucky and have QS fall early on someone else, sluff most of this suit, Shoot the Moon, or prepare yourself to meet the Lady. If you hold QS your problem is different, especially if your holding can withhold a siege of spade pulling. There is freedom in holding QS with a decently long spade suit! You alone can play without fear of taking her on a non-spade trick. Freed from the anxiety that afflicts everyone else at the table, you can play as high as you wish. And you should play high early and often. Your goal is to build voids and eliminate entries into your hand, so that you can aim QS at a target with imperviousness. It is much more important to achieve this attack position than it is to mask the fact that you hold QS. Sooner or later every Hearts player confronts this philosophical question deep into a hand: What the hell do I lead now? "I have this here, li'l ol' 2D. Is it an out-card, or is it, by any chance, the 13th diamond?" It is amazing how many times a deuce or trey in a dead suit is led late in a hand and is awarded QS! So this concept should be tacked on the inside back wall of every serious Hearts player's skull: *** Count the cards! *** Keep a running count of how many cards have been played in each suit that you hold. For extra credit, count all the suits when you hold QS. This can help you put her on your target. Only those blessed with exquisite memory will also be able to remember which cards the opponents still hold. Memory is the secret weapon of Hearts. Remember the critical things. If you are treasuring a 3C as an out-card, remember whether the 2C has been played. If it has not, your 3C may not be an out-card! If you hold QS, remember if the AS or KS has fallen. It sometimes happens that you hold QS and realize that a target holds the only spade still out, a KS or AS. Few plays in Hearts are more dramatic or intimidating than leading QS and causing the AS or KS to drop (this is something that Hearts-playing programs do far better than people)! Remember the suits in which your opponents are void. The player who sluffs after you played high on an earlier trick in this suit either does not have QS, was targeting someone else, or was Shooting. So, leading this suit is relatively safe: It may get you out of the lead, and it may cause QS to fall on the low scorer. At worst it should cost no more than a heart or two. A good memory is essential when you have the lead late in a hand and QS has not yet fallen. No holding in Hearts is more precious than an out-card late in a hand! The truest out-cards are, by definition, the lowest unplayed cards in the live suits. Other than a low spade, you have only your memory to determine if you have an out-card. Never squander an out-card. Hearts offers the opportunity to develop many subtle strategies. Nevertheless, I repeat that everyone will take many turns dancing with the Lady! You will be doing well to win a third of your games. Bad hands happen, sometimes with painful regularity. I have been dealt a singleton AS, KS, or QS on a Hold hand more times than mere chance would explain! Do not bellyache when it is your turn. And when you drop her on a player whom you have been chasing all evening, do not gloat. The Hearts player's mantra should be, Gloat not that ye be not gloated over! Cackle now and pick her out of your ear in a moment! Did I mention that good Hearts players have good memories? Playing Mode: Low-Scorer Being low-scorer means being the focus of everyone else's attention, for they envy your position and will do anything to displace you. As a wise person once observed, you are not paranoid if they really are out to get you. This brings us to the concept sometimes inelegantly referred to as "lying low." The idea is for a player to build a "hole" to hide in and then disappear into it, pulling the hole in after her and taking no more points. Lying low is largely the art of building voids. A big part of lying low is not going high (hence the term) on a non-spade trick, unless you are sitting last, are playing a Hold hand, or the player who holds QS has already played on the trick. As already discussed, making non-hearts voids is the most important strategy in playing Hearts, and the earlier you have a void, the more useful it is. So, if you have the lead early in the game, consider playing your singletons and doubletons. Use your void to sluff problem cards and to create another void. Break hearts, if hearts is a good suit for you (i.e., you have few hearts or many low ones), or you are concerned about an opponent Shooting. Certainly split hearts, if one player has all the points. If you are in trouble holding QS, as low scorer you should dump her at your first opportunity. If you do not hold QS, generally your best strategy is to lead spades whenever you can, unless you are in trouble holding the AS or KS. Your main targets are the high-scorer, to nudge her closer to the finish line, and your nearest competitor, to widen your lead. Playing Mode: Lying Low This mode of play is appropriate when your score is close to low. Here you keep your eyes down and try not to be noticed. The theory behind this mode of play is that sooner or later the low-scorer, who is everyone's target, will get nailed and you will silently slip into first place. If you have QS, aim her at a needy opponent but not on a player who will go out (reach 100 points) causing you to lose! If you note that the low scorer has had the opportunity to lead spades but has not, you might suspect that she has spade trouble. Then you should consider pulling spades, unless the low scorer sits last on the trick and the AS or KS has not been played (if the low scorer sits last and you lead spades, you may help her out of a tight spot by enabling her to get rid of a naked AS or KS; or you may inadvertently pull another opponent's AS or KS, causing her to win the low-scorer's QS). Some tricks are worth coming out of seclusion to take, such as the trick with the first heart. This guarantees that no one can Shoot the Moon but you. Always take a heart-bearing trick if another player has all the points. Taking a few hearts is usually cheap insurance! Another trick to take early is one in a suit in which you have a high-low doubleton. Take high, lead back low. Presto! a void. Playing Mode: Getting Low-Scorer The new player dumps QS, the expert aims her. Hearts has only one winner and three losers - second place does not count! Even though Hearts is a cutthroat game, experienced players learn to play cooperatively. Late in a game the three high scoring players form a loosely knit, unofficial, unspoken coalition with just one purpose: to damage the low scorer. Being able to lay points on the low-scorer is probably the most important (and rarest) skill in Hearts. If a player cannot do this well, once she falls behind she can only hope that the low-scorer self-destructs, that she can overtake low-scorer by Shooting the Moon, or that some another opponent has the skill to undo her. Since almost everyone wins less than half the games, every Hearts players spends most of her time in this mode. The essence of playing well in this mode is patience and fortitude. If you hold QS and it is not in immediate danger, you must pass up sluffing QS on other targets that may present themselves and wait for low-scorer to take a trick. Try to lead a card that will cause low-scorer to win the trick. This means paying close attention to the suits in which low-scorer is void and being very careful not to lead them. If you cannot lead into low-scorer's hand, perhaps you can lead into the hand of an opponent who sits to low-scorer's right, so that she can then lead into low-scorer's hand. On a pass-across or pass-right hand, it may be worth considering whether to pass QS plus a spade or two to the opponent who sits directly behind low-scorer. This player has the best chance to christen a trick on which low-scorer leads high. Playing Mode: Shooting the Moon Shooting the Moon is like jujitsu: it uses your opponents' strengths against them. Hearts players learn to avoid points. All a Shooter must do is let the other players do what they do best, duck points and discard winners, while collecting all the points herself. I told you Hearts is a simple game! What is essential to Shoot the Moon? Clearly, lots of high cards. Although this can be achieved in all four suits, it is usually easier if you have one or two long, strong suits. You must have very few losers after you have taken some hearts and the other players are beginning to awaken to your intention. You must have the ability to play coolly while counting cards in a pressure situation. And, most importantly, you must have the nerve to try. Score should be a factor in your decision. If your score is in the stratosphere and you are 20 or more points behind, you are running out of time and will probably need to Shoot the Moon to win. Because you are in serious scoring trouble, it may be risk-appropriate to try to Shoot with a hand that is not as strong as you would like. If your opponents are serious, experienced players, they will help you, as we have discussed - especially if your Shoot's failing will cause them to lose the game! On the other hand, if you are winning by a lot, the benefit of Shooting does not equal its risk unless your hand is extremely strong. If you are considering Shooting the Moon, you will probably have to pass away losers that you would otherwise cherish. At worst your pass may reveal to one player that you are thinking about Shooting what else is she to make of receiving 2D, 3D, and 4D! You must also be able to assimilate whatever pass you receive. If the pass you get kills the Shoot (e.g., 2H, 3H, and 4H into your heart void), then cancel the Shoot without delay. Play your losers early. Use losers to draw out your opponents' winners, promoting the power of your other cards. However, you need not lead all your losers. That is, you can save a loser or two (especially if QS has not yet been played) until the last tricks, forcing your opponents to guess the suit of your weakness. Your losers are considerably strengthened if you also hold QS, for few opponents will risk trying to stop your Shoot at the cost of winning QS themselves. Once you commit to Shooting, you must take all the points quickly or plan to take all the point-bearing tricks. A long, strong heart suit is easiest on the blood pressure, because you can drain all the hearts in a few pulls. Is it possible to Shoot the Moon on a Hold hand? You bet it is! Playing Mode: Equivocating about Shooting the Moon The best Shooters hedge their bets by thinking offensively and defensively at once, before they commit outright to Shooting or decide not to try. Most hands that produce successful Shoots are not ironclad from the outset. In fact, they may have severe weaknesses that, through skill or good fortune, are overcome. When equivocating, one plays losers on tricks without points, gradually building up the hand's strength in its remaining cards. The skill is in being able to prune losers without having to commit to Shooting and without being overly punished for one's boldness. Hands with one or more long suits lend themselves especially well to equivocating. A long suit often has out-cards at the bottom end with which to Lie Low if a heart should fall to an opponent. Playing Mode: Stopping a Moon Shoot The defense is usually at a disadvantage when someone tries to Shoot the Moon. Because a serious Shoot attempt generally occurs less than once a game, its infrequency can mask its presence. It may take several tricks before their common problem dawns on the others. Most Hearts players develop fine defensive skills, because that mode dominates Hearts play. Stopping a Shoot requires taking a trick and seeking a point, an entirely different mindset. The best defense against a Shoot is not the pass, which can only hold one opponent in check, as we have discussed. Rather, the best defense against a Shoot is an early void, which can be effective against all opponents. A sluffed heart has aborted many a Moon Shoot, leaving it smoking on the launchpad. There is an interesting logical progression here: 1) The earlier a player develops a void, then 2) The more tricks the Shooter has to control, then 3) The stronger the Shooter's hand must be, then 4) The less often that Shoot-worthy hands occur, then 5) The less often the Shoots, then 6) The less often that opponents anticipate a Shoot, then 7) The more likely Shoots are to succeed. As mentioned, skillful players do not wait for a perfect hand to Shoot the Moon. Many a Shoot has developed from a hand that originally contained several losers. The problems for the defense are to figure out in which suit is the Shooter's loser, then to beat it by taking at least one point but as few points as possible. The Shoot succeeds when the Shooter captures all the points. Therefore, as long as the Shooter has not captured some points, the Shoot may be problematic. The defensive player must guess correctly in which suit the Shooter has a problem and retain a winner in it. Someone else must retain a point card. Only one player has to guess right if the Shooter's weak suit is hearts. However, because the point cards that defenders most often retain are their highest hearts, it is unusual for the Shooter's weak suit to be hearts. But what if QS has not fallen and you hold the AS, KS, or QS. Do you really want to stop the Shoot that much? Determining the Shooter's weak suit is a problem that memory helps. Remember in which suits the Shooter is void, for clearly she has no loser in those. Remember to sluff your dead cards except in hearts, where you might want to retain one or two. Merely taking a trick does not stop a Shoot; it must contain a point card. Playing Mode: Helping High-Scorer Shoot the Moon The three players who are not low-scorer form a cooperative coalition which is dynamic and unstable. Its members accept that just two things bind them together: * Each wants to beat the hell out of the others, and * The low scorer is currently thwarting them all Unspoken is each member's knowledge that she has a better opportunity to win if only the low scorer can be dislodged. It matters not that, if the coalition is successful, one of its members will be the target of a new coalition. Is an opponent successfully Shooting the Moon always a bad thing? No. If the Shooter's score is more than 26 points higher than yours, it may be better to let her Shoot than to run the risk of stopping her. Consider also the case in which you are not low-scorer, the Shooter's score is near 100 points, and her score is more than 26 points higher than low-scorer's. A Hearts game is over when one player reaches 100 points. It follows, therefore, that a Hearts game in which no one reaches 100 points continues forever! Recall that a successful Shooter may deduct 26 points from her score. A Hearts game in which high-scorer pulls back from the brink gives new life to the game, for a game that is still being played is one that anyone can win. Considering that you lose if you do not win, a high scorer who decides to Shoot and cannot win the game by doing so should be able to count on two supporters and one enemy (low-scorer, who wants the game to end now). But even low-scorer will not be interested in stopping this Shoot if it risks dumping her out of the lead. The Shooter's supporters (who may even include low-scorer) discard their winners before these cards can capture tricks. They allow the Shooter to capture quickly their hearts and QS, if they have her. Remember that Shooting the Moon does not require taking all the tricks, just all the points. Playing Mode: Protecting High-Scorer Playing cooperatively is an enlightened broadening of the definition of what constitutes one's best self-interest. It reaches a high form when a player refrains from dropping QS on a member of the coalition (even second-place player), deciding to wait for the coalition to maneuver into a position where she can give it instead to low-scorer. Her fellow coalition members, who understand (and expect) that she will restrain herself in playing QS, reciprocate by not leading spades unduly, especially when one knows that the only spade remaining outside her hand is QS. The challenge is to get the lead and lead low in a suit in which the holder of QS is void and low-scorer holds a winner. Here, again, memory and the fall of the cards are key. Cooperative play can be painful. Consider a trick on which you play last, and you are not low-scorer. A 5D lead drops low-scorer's QS. A player whose score is 88 has to play 7D. You have 4D and JD. If you follow your instinct and duck with 4D, high-scorer takes the trick. Congratulations, you have just shot yourself in the foot, for the game is over and you lose! The only way to keep the game, and your chances to win it, alive is to groan loudly, play JD, and win QS. Another painful situation is when a protecting player knows that the only spade still remaining outside her hand is QS, knows that high-scorer holds QS, and realizes that her hand has no other outcard except in spades. If she leads a spade: game over and she loses! She must lead another card and probably wind up eating QS! But she will live to play another hand, and where there is life there are hope and possibility. It is not unknown for a protected high-scorer to Shoot the Moon and win the game! Few strategies are risk-free. To watch three excellent Hearts players playing cooperatively, stalking a low-scorer adept at evading, is to watch Hearts at its very best. Musings Hearts has a high chaos factor. The chaos is due to the lack of information about other hands, especially concerning skewed suit distributions early in play and the whereabouts of QS. Contrast this with Checkers and Chess, two games with zero chaos factor because everything is visible and, therefore, knowable. An inherently high chaos level tends to diminish the effect of skill. A corollary is that poorer players do better against good players in Hearts than in many other, less chaotic strategy games. One definition of a poorer player is one who brings randomness (in the form of lack of knowledge) to a game. The effect of a poorer player's randomness is masked in a game with an already high level of chaos. The author of another Hearts-playing computer program has written me that he was once trounced by three computer players who played by the rules but picked cards at random. If in Killer Hearts a Best Player plays three Random Players in a set of 5,000 duplicate games, Best Player will probably not win them all. Still, playing with better players is more challenging and more fun than playing with poorer players, and there is a big difference in the feel of the game. Skill tends to make order out of chaos. As in other, more serious realms, intelligence tends to reduce and can even reverse entropy. It follows that the skill of a good player makes your own skill more important. And yes, although it may seem strange to say so, it is more satisfying to receive QS by design rather than by accident, by aim rather than by dump! Much of the chaos in Hearts is due to the three-card pass, because it so drastically skews suit distributions. It is interesting, however, that most experienced players (including me) favor passing hands over Hold hands. Each believes that she can use the pass to improve her hand, despite the pass coming her way. Where does the skill lie in Hearts? It lies in being able to nail a proper target with QS. In playing cooperatively, because this prolongs the game, thereby increasing winning opportunities for the three players who are not in the lead. In determining where QS sits and playing accordingly. In passing well. In playing well with a bad hand. And in evading skillful efforts to stick you with QS. Shooting the Moon is less a skill than a capacity to accept risk (or recognize opportunity) and the boldness to exploit the defensive nature of the game. But being able to stop a Shoot consistently is a precious skill. A Little Math and Other Fun Things The best Hearts player does not always win. In fact, she may not even usually win. Over a large number of games between the same players, the best player will win the most games (this is what makes her best). But lacking a large number of games and mindful that low scores win games, can we say that the best player is the one who takes the fewest points? Moreover, is a player's scoring average (her proportion of all points taken) an accurate predictor of her long term winning average? It might seem reasonable to think so. But the somewhat surprising answers are that the lowest scoring average may not win the most games, and that scoring average is not a reliable predictor of winning average. Since, in Hearts, the lower the score the better, the proposition that winning average is inversely proportional to scoring average is expressed by W1 is proportional to 1/p1 where W1 is player one's predicted winning average and p1 is her actual scoring average (the percentage of all points that she takes). If the scoring averages are firm and knowing that p1 + p2 + p3 + p4 = 1 then W1 = 1 / (p1 * (1/p1 + 1/p2 + 1/p3 + 1/p4)) Now consider Table 3, which contains scores from ten consecutive Hearts games. Player 1 is a person and players 2, 3, and 4 are identical replications of another computer game's best player. Clearly player 1 has won many more games (8) than his scoring average predicts (.342), and the automata have won fewer. This trend continued over the much larger set of games from which these ten are extracted. Over 100 consecutive games, player 1's predicted win average, W1, was .316, but his actual winning average was .550. He won 55 games instead of the 32 that his scoring average predicted. That is 23 games and 74% more than predicted! What goes on here? Player 1 Player 2 Player 3 Player 4 Scores in Game 1 62 101 69 80 Scores in Game 2 94 96 88 112 Scores in Game 3 64 94 67 113 Scores in Game 4 6 118 20 12 Scores in Game 5 17 101 65 77 Scores in Game 6 102 75 83 78 Scores in Game 7 51 62 55 118 Scores in Game 8 44 65 92 111 Scores in Game 9 34 117 89 98 Scores in Game 10 57 68 87 100 Total Points 531 897 715 899 Scoring Average, p .175 .295 .235 .296 Predicted Win Avg., W .342 .202 .254 .202 Actual Wins 8 1 1 0 Actual Win Average .800 .100 .100 .000 Table 3. Data from Ten Consecutive Games. The situation can illuminated by example. Consider a hand in progress in which players 1, 2, 3, and 4 have 64, 96, 54, and 111 points respectively. When we peer into this hand, all the hearts have been played and split, player 1 holds QS with no entries into his hand, and another player has the lead. Clearly player 1 is about to put QS on someone. Player 1 only wins the game if he gives QS to player 3, but his scoring average will be the same no matter who his victim is .189 (64/338) and his predicted W for this one game will only vary from .297 to .313! Part of what we are seeing is the value of being able to put QS where it will do the most good. If player 1 merely drops QS at random, he wins fewer games than if he aims QS at an appropriate victim but his scoring average is unaffected by his aim! Player 1's ability to aim QS at the proper target increased his winning average and helped enable him to win 23 games that he might have lost with the identical scoring average. So scoring average is not a reliable predictor of winning average. Your scoring average in part reflects your ability to evade QS and other points. But that is only the defensive aspect of Hearts. Your ability to win is also very much a matter of your offensive ability to aim QS what we may call "queensmanship"! Playing cooperatively also skews the predictive ability of scoring average away from winning average. A player may take points that she did not have to, were she not trying to keep the high scorer from going out and, therefore, the game and her chances to win it alive. Furthermore, a player who often tries to Shoot tends to have a higher scoring average than otherwise, but if her Shoot success rate is high enough her winning average will increase. Does this mean that a player can win more games than someone whose scoring average is significantly lower? Yes! For a reductio ad absurdum case showing that scoring average is decoupled from winning average, consider these four new players. Player 5 plays successive games in which she scores 112, 111, 5, 112, 111, ... points. In those same games player 6 gets 111, 5, 112, 111, 5, ... points, while player 7 receives 5, 112, 111, 5, 112, ... points. Meanwhile, player 8 takes just 6 points in every game. Players 5, 6, and 7 have scoring averages of .325 but a winning average of .333. Player 8 has the phenomenally low scoring average of .026, but she never wins a game! In the real world, however, where such scoring sequences are highly unlikely, the best player of a group usually does have the lowest scoring average. However, a somewhat better indicator of quality of play than scoring average is each player's proportion of the number of points in excess of the winning score per game (the ratio of a player's average losing margin to all losing margins). This measure (call it D) takes into account a player's ability to aim QS. If the scores of a game are 106, 64, 34, and 82, then the total number of points in excess of the winning score for all players is 150 (72 + 30 + 0 + 48), and D1, D2, D3, and D4 are .480, .200, 0, and .320 for this one game. The relationship between D and winning average looks familiar W1 = 1 / (D1 * (1/D1 + 1/D2 + 1/D3 + 1/D4)) Using this equation to predict winning averages from the scores shown in Table 3 yields .759, .063, .115, and .062 for players 1, 2, 3, and 4. This correlates quite well with the actual small sample (.800, .100, .100, .000). Having examined the results of this predictor of W over large numbers of games, I find that it almost always provides a better estimate of winning averages than an estimate based on scoring averages. However, it tends to underestimate higher winning averages and overestimate lower winning averages. It, too, must be considered unreliable. Do players with the same skill level have the same scoring and winning averages? I have studied at length Killer Hearts and another Hearts-playing program, always playing against three replications of each program's best player. Clearly, identical replicants have identical skills. After hundreds of games, I have found that the answer is no. The player playing after me did best in both scoring and winning averages, the player at my right did worst, and the player sitting across from me fell between them. These data suggest that the best seat at the Hearts table is behind (to the left of) the best of the other three players. For something different, consider this. Can a game end with all four players having 100 or more points? Yes. Final scores of 106, 104, 103, 103; 107, 104, 104, 101; etc. are possible. To get to the first set, the scores before the last deal could have been 93, 99, 99, 99. The highest possible score in Hearts is 125, but what is the highest score with all four players at 100 or more? It is 116 (the scores are 116, 100, 100, 100). Is a four-way tie at a score above 100 possible? No. The total of points taken in a game must be a multiple of 26. Consider the possibility of a tie at 104 (4 x 104 = 416 = 16 x 26). To get to this score, the player who took QS on the last hand could have had no more than 91 points at the start of the hand. That leaves 13 points for the others, but 15 are needed for the other three players to have had scores of 99. The next multiple of 26 is 442, which is not a multiple of 4. Next is 468, but a tie at 117 is not possible; three players would have had scores of at least 109 at the start of the last hand. A player who Shoots the Moon has the option of reducing her score by 26 points. Is it possible that a rational player's score can go below zero? No. The only case in which it behooves the successful Shooter to reduce her score, rather than adding 26 points to everyone else s score, is when her score before the Shoot was 26 or more points higher than the low score. So the lowest her score can be before the Shoot is 26 and the lowest it can be afterward is zero. The shortest possible complete Hearts game requires four deals. The shortest games I have ever seen were over in six (for an example, see game 4 in Table 3). The longest game without a successful Shoot consumes 16 deals, but I have never seen a non-Shoot game exceed 15 deals. The rarest events in a game of Hearts? A player taking no points, a player scoring 125 points, all four players scoring 100 or more points, a margin between first and second place of more than 75 points, and one player successfully Shooting the Moon on three consecutive deals, to name some. (I have seen one each of a winner having zero points, a high score of 124 points, three players over 100, 77 points between first and second place, and two straight Moon Shoots with the same Shooter finally failing on the 13th trick of the third game.) Afterword Hearts is a game in which skill will out eventually, but maybe not today. For this reason Hearts tests your capacity to endure adversity, frustration, defeat, and even humiliation. It makes you sweat and wonder what you did to offend the deity. It makes you think and work at improving your memory. It teaches: Cooperate or lose! Narcissistic or arrogant people should avoid Hearts unless they are very, very skilled. Even so they will still be smashed, as we all are. Hearts is a game that begs not to be taken too seriously. Fun is its main product. Fortunes are not made or lost on it. People do not write songs about it. There are no systems for beating it. Master points are not awarded for excelling at it. It is a game that is meant to be played and enjoyed. But having both won and lost, I know that you will enjoy Hearts more if you win more. It pleases me to think that I may have helped you do that. [Early versions of this paper, which I wrote many years before developing Killer Hearts, helped greatly in designing Killer Hearts.]