Sure Things

Death and taxes have long been recognized as the only sure things worthy of a cautious man's faith or wager, and we know some individuals who are suspicious of these. (There is a movement afoot to add the Green Bay Packers to the list, but that seems a trifle premature.) Yet a considerable number of other propositions have outcomes so certain that they warrant the interest of even the most cautious of men. These are "sure things," and they result in gain for the initiate by causing his ill-informed prey to become intrigued—and indebted. Not ruined or overdrawn at the bank, however, for these are gentlemanly swindles meant for rewarding diversion rather than malevolence.

Our purpose here is to present some of these entertainments as a doubly beneficent public service: The extroverted reader will make immediate use of them; they are a perfect pastime while you're waiting in some lounge for a late plane, train or date and the time needs to be whiled away. In fact, they'll do at any moment when one is not precisely where one would like to be and the conversation is likewise not what it might be. In the future, those moments can be spent in the pleasurable pursuit of profit. And the introverted reader who might be mistaken for a "mark" will now be one up when some aggressively friendly fellow just happens to offer a little wager to help in the whiling away of that same dull moment.

On the safe assumption that there is some larceny in all of us, our diversions are presented as the "operator" needs to know them. One note of advice: Only the bare mechanics are outlined here. In order to ensure a long and lucrative career, you must be able to awaken and entice the avarice of your prey. This does not imply, however, that you need the pitchman patter and ingratiations of the stereotyped bunko artist. In fact, the most successful operators we know are both quiet and somewhat diffident in disposition, prodding only when necessary and easing up once the barest response is evident. The one compulsory trait is to demonstrate good-natured interest in the proceedings, as if there were really a game of chance under way.

For the skeptics and slow learners among you, a detailed explanation of each ploy has been appended to our list.

Now for the games.

1. A mathematical oddity called Crazy Eights. A pencil and paper are necessary; they are for the pigeon. In a charmingly straightforward way, you ask him to pick a number. Then, in order, he is to double it; add 25; square it; and fold up the paper. Now, you tell him—after seeming to make some sort of computations in your head as he did them on paper—if he subtracts 25 from his final number, it will be divisible by 8. This should elicit a response from your companion. He probably does not even remember the number he's computed, so there'll be an inclination to protest your arithmetical arrogance. When it comes, offer a small wager; if necessary, give odds. His number will divide quite nicely by 8—even if he cheats; this bet never loses.

2. Instant Math. You should now have at least an interested and possibly an angry prey (the latter is a definite advantage: The angrier he gets, the more susceptible he is, ultimately becoming an abject sucker). It is time to bring him along with another example of your mathematical wizardry. Calmly state that you've mastered the 15,873 multiplication table. Your opponent will be wary, but he'll register "Show me" in some subtle way. "That's crazy!" he might say. So you ask him to pick a number from 1 to 9. Tell him you're going to participate by doing the same, whereupon you write the number 7 on a piece of paper. Then you offer to multiply the 7 by his number by 15,873 within 3 seconds. This feat, certainly, is worth a wager. After the stakes are set, you ask him for his number and proceed to write it down 6 times. (For example, he picks 6: 6 times 7 times 15,873 equals 666,666.) The cloak of infallibility can be seen settling comfortably upon your shoulders. (continued on page 174)

Sure Things

(continued from page 111)

3. Our third diversion involves math more deviously. It is called Thirty-One and is a variation of a famous game called Nim that was immortalized as the Match Game in the film Last Year at Marienbad. In this version, you place on a table 31 matches. You explain that each of you must take turns picking up at least 1 but not more than 5 of them. The picker of the last match loses. You invite your guest to go first. You win this game by thinking in multiples of 6. Each time your pigeon picks, you take a number of matches that will make his turn plus yours equal 6. Obviously, after five turns, the last match is his.

Your officious etiquette in allowing your opponent to start each time will raise some suspicion and he will probably at some point invite you to go first. When that happens, you take any number, watch his move, and then be sure your next pick makes a grand total of either 6 or 12, whichever is available. You will be in the same position as earlier and make groups of 6 to the end.

If you find that he is making 6s after your first pickup, so that you cannot, it's time to switch games, for he's caught on. If your opponent insists from the outset that you go first, he probably knows the game and you'd best demur: He will not grant a rematch. Obviously, if the game goes normally, there are many opportunities for betting, depending on your mood.

4. Salaries. Our next maneuver is useful when one is cloistered with an acquaintance who marvels at his own financial abilities. You take this genius into your confidence and tell him you've been troubled by an important financial decision. A prospective employer has offered you a choice of two methods for receiving salary increments. You may receive either a $250 raise every six months or a $1000 raise each year, and you, poor simpleton, don't know what to do. You can depend on at least a patronizing pat on the shoulder and the fatherly advice to take the $1000. But that's a stupid and costly thing to do, you say. His likely rejoinder will be something subtle, such as, "Oh, yeah, you wanna bet?"

After negotiating the stakes, you explain: We'll figure the raises on a base pay of $1000 per month. If you take the $250 raise every six months, then you earn $6000 after the first six months, then $6250 after the second six months: a total of $12,250 for the first year. The $1000 option offers no raise until after the first year, so the salary is only $12,000.

Now it seems that in the second year the $1000-per-year option would do better. After all, there'd be $13,000, while the other gets only two $250 raises. But a closer look shows that the second $250 raise again comes after six months, bringing the total for the half year to $6500 ($1000 monthly base plus two raises), and the next raise at year's end brings that half year's pay to $6750, a total of $13,250. From then on, the two half-year raises always produce .$250 more per year. Simple? One note of caution: Don't pose this one to your boss the month you're up for a salary review.

5. There are occasions when it helps to have a situation in which your victim can function alone, so you can demonstrate that it's his failure of character rather than your cunning that is costing him money. A useful situation of this sort is presented by a game we call Utilities. You show the following diagram:

You explain to your companion that he has just become a real-estate developer (that should help his ego tremendously). But he has a problem. He has completed building the charming $100,000 Colonial homes shown just as the gas, water and electric companies start a feud. Each of the companies refuses to allow any of the others to install lines across its own. Now he's got to solve the problem by drawing a line connecting each house with each utility, without any of the lines crossing one another. When a response is elicited, offer any odds you like. The problem is insoluble.

6. If the pigeon was too shrewd to take your bet on Utilities, and he fancies himself something of an engineering or mathematical marvel, we'd like to present a diversion designed for him. It's called Excavation. The problem is that you have drilled a hole through the center of a sphere. You measure and find that the hole is 6 inches long. What is the volume of the remainder of the sphere after drilling the hole?

Now, it's possible that your prospect will immediately give you the answer. If so, he is indeed the bright fellow he credits himself as being, or he read this before you did. Either way, thank him and buy a round of drinks. More likely, however, he'll ponder the problem for a while and inform you that you've made a mistake—the problem can't be solved. He'll ask for more information, such as the size of the sphere or the diameter of the hole. You assure him that there is enough information and, if you are of such a bent, make derogatory references to his vaunted mathematical acumen. Obviously, it is time for the wager.

The solution to the problem is always the same: the square of the length of the hole times pi cubic inches (or feet, etc.); in this case, 36 pi cubic inches. (You can multiply it—36 times 3.14—if you've a feeling for verisimilitude.) You may make it all look difficult by doodling arcane symbols and figures for a while before springing the answer, with lots of sighing and brow knitting—the stylistic embellishments are up to you. But the answer is no problem, and we'll give you an unimpeachable source in the explanations that follow.

A problem may arise with your prospect because all of the above are so absolutely foolproof. Even the most self-destructive sucker gets impatient when he realizes that there's no hope at all. It's now time to introduce him to some entertainments where he wins just often enough to stay interested, while playing long enough to plentifully reimburse you for your time.

7. The first of these more conventional games of chance is Three Dice. You ask the prospect what he thinks the chances are of rolling at least one 6 with three dice. Your average pigeon will quickly calculate that there's one chance in six with one die; therefore, there must be three chances with the three dice. It seems to be an even-money bet. Actually, however, the odds are about 4–3 against a 6 turning up. If the numbers have no meaning for you, don't fret. Just remember that they are considerably better odds than the casino at Monte Carlo uses to accumulate rather large sums.

You now have a number of opportunities. You may simply offer even-money bets against 6s and steadily increase your cash reserves. If you're in a dramatic mood, you might launch into a soliloquy on your occult powers, ending with a pronouncement of telekinetic prowess. You offer to demonstrate these by assuring your mark that you can prevent his rolling 6s; and you'll show just how much faith you have in yourself by placing some gentlemanly wagers.

If signs of boredom set in, offer to pay double when two 6s show, triple for three. Under the new system, the odds are still a comfortable 11–9, your favor; and the latter appeal should substantially lengthen your pigeon's attention span.

8. Similar to the above—and no less profitable—is Triplets, a brutally simple money game whose action is faster than a Las Vegas crap table. Three coins are needed, plus a sucker. The coins are all tossed at once. If they come up three heads, the sucker gets $10. If three tails, the sucker gets $10. Any other combination and he pays you $5. That's two wins for him out of three possibilities, plus odds. Sounds too good. But if you con him into playing, you win three times out of four. That's $15–$10, or a fast $5 take in a very few seconds. Played over long periods of time, this one loses friends and turns acquaintances into solid enemies.

9. A gentler game is Queens. Take two kings, two queens and two jacks from a deck of cards. Turn them face down on the table and shuffle so neither of you knows which is which. Chatter amicably about how unbeatable your companion is with the fair sex. Then tell him that he's so magnetic, if he picks two of the cards, one will be a queen. Make a bet: it's 3–2 he's got a queen. If he objects because there are only six cards to choose from, offer 10–1 odds that he can't pick both queens. That's a very sweet bet. The odds are actually 15–1 against it.

10. Two of a Suit. We're going to end our lesson with one of the simplest and most deceptively effective of these bets. Take an ordinary deck of playing cards and have your prospect cut them into three piles. You propose that when one card from each pile is turned up, two of them will be of the same suit. As usual, you are willing to back your proposition with hard cash. The most unlikely people will call this bet; the proposition truly seems foolish. In fact, dear reader, you might seriously ask yourself which side of this bet you'd be inclined to take. After all, there are four suits, three cards and an honest deck. But speculation doesn't phase the odds: They are, in fact, slightly better than 3–2 in favor of getting two of the same suit from a random pick of three cards—once again, a bet designed to turn a nice long-term profit. Why not try it out right now? It's an enlightening experience, and the first step of an entertaining avocation as a gentleman swindler. Bonne chance!

The Explanations:

1. Crazy Eights. This bet takes advantage of the fact that any time you square an odd number and divide it by 8, you get a remainder of 1. The steps of the bet set up the right situation. When the mark doubles his number, he makes sure it is even at that point. He adds 25 to make sure his number is odd at the next point. Then he squares it; at which point, if he divided by 8, he would get a remainder of 1. Then he subtracts 25—which is 3 times 8 plus 1—which gets rid of the remainder and assures the operator of a win. The exact numbers used are arbitrary: Any odd number would do instead of the first 25, and 1 or any multiple of 8 plus 1 would do instead of the second 25. In effect, you could just as well have told the sucker to take an odd number other than 1, square it. and then offer him odds that if he divided by 8, he would get a remainder of 1. But the elaborate version provides the necessary drama. The proof that the square of an odd number always leaves a remainder of 1 when divided by 8 is fairly simple and is left as an exercise for the reader.

2. Instant Math. The explanation here is trivial. Obviously, if you multiply 111,111 by any digit, say 5, you get a string of 6 of those digits—in this case, 555,555. And 111,111 divided by 7 gives you 15,873. So, of course, if you multiply 15.873 by the pigeon's number and then by 7, you obviously get ... his money.

3. Thirty-One. The formula was stated in the description of the game. There is nothing special about 31 matches or a maximum pick of 5. If there were, say, 50 matches, with a maximum pick of 7, the operator would divide 50 by 8 (that is, by groups of 1 match more than the most the opponent can pick up at a turn). He sees there are six groups of 8 matches, plus 2 left over. He offers to go first, picks up 1 match, and the sure win is established. After six more turns, there is an odd match left for the loser.

4. Salaries. The gimmick is just that the two raises are not being calculated on the same basis. The $1000 raises not only come but once a year but are on an annual rate basis; the $250 raises are being calculated on a six-month basis, which comes to $500 per year for each $250 raise. So the question is really whether or not a $1000 raise is better than two $500 raises. If you put it that way, then it is pretty obvious that the "smaller" raise is better, since it amounts to the same thing annually as the bigger raise, except that you start getting paid a higher rate sooner than if you had to wait till the end of the year. But, of course, if you put it that way, you don't have a bet.

5. Utilities. The proof that this problem can't be solved is not too hard to understand, but it takes more space than is available here. Consult any elementary text on topology. Of course, you don't need proof to win the bet.

6. Excavation. The reason the answer is always the same, no matter how big the sphere was, is that in order for the hole to be exactly 6 inches long, it has to get wider and wider as the sphere gets bigger. It might help to think of the limiting cases: When the sphere was only 6 inches across (i.e., its radius was 3 inches), the hole through the center must he infinitely small; it is just a line 6 inches long, with no volume. The volume left is the whole original sphere, which is 36 pi cubic inches. On the other hand, as the size of the sphere approaches infinity, the space between a line 6 inches long and the side of the sphere approaches zero. It never gets to zero, of course; there is always enough left to get your remaining volume of 36 pi cubic inches. If you are interested enough to go to a library, you can find the mathematics in the November and December, 1957, issues of Scientific American.

7. Three Dice. The probability of not rolling at least one 6 with three dice is simply 5/6 times 5/6 times 5/6, which is 125/216, or about 58 percent. The 6 comes up the other 91/216 times, or about 42 percent. A layman's approach of adding the probabilities and getting 1/6 plus 1/6 plus 1/6 suggests that the chance is 3/6, or 50 percent, which seems reasonable and is simpler, but wrong.

8. Triplets. The chance of getting one head, if you toss one coin, is, of course, 1/2. The chances of tossing three heads with three coins is just 1/2 times 1/2 times 1/2, or a net of 1/8. Same for tails. So the mark wins a total of 2/8 of the time, and the operator wins the other 6/8 of the time, enough to make a comfortable profit even after giving the 2–1 odds.

9. Queens. There are six cards, two of them queens. This makes the odds 4/6 against his getting a queen on his first pick. If he does get a queen, the operator has already won. If he doesn't, there are five cards left for the second pick, of which two are still queens. His chances of not getting a queen the second time will then be 3/5. The over-all odds of not getting one queen are 4/6 times 3/5, which reduces to 6/15. The operator wins the other 9/15 of the time, which is a nice situation to be betting even money. On the other bet to get both queens (at 10–1 odds), he has 2 chances out of 6 to get a queen on his first pick. If he doesn't get a queen, he has already lost. If he does, his chance to get the remaining queen on the second pick is 1 out of 5. So his over-all chance is 2/6 limes 1/5, which reduces to 1 chance out of 15. The operator wins the other 14 times.

10. Two of a Suit. The reason this bet sounds so attractive is the tendency to confuse the situation where the operator has to get two of a particular suit, say spades (where the odds would be very much the opposite), with the one here, where the sucker loses whenever he fails to pick three different suits. Put that way, the bet doesn't sound very templing at all, which is why the operator never puts it that way. (Think of the cards being turned up one at a time: After the second card, either the operator will have already won—because the first two cards were of the same suit—or he has a 50/50 chance of winning on the third, where two suits will win for him, and the other two will lose. So the operator wins about 1/4 of the time on the second card, and the rest of the lime he has another chance—and a good chance—to win on the third card. Your guest wins what's left, which is not enough to allow any but the very rich to play this game for long.)