L.A. Math: Romance, Crime, and Mathematics in the City of Angels

by James D. Stein

  The next day was New Year’s Eve. At about ten in the morning, the doorbell rang. It was a messenger bearing an envelope addressed to Pete. Pete signed for it, lighting up like a kid on Christmas morning. He opened the envelope. The first item I saw was a season ticket to the next year’s Raiders games, exhibition games and playoffs included—also parking, which at an L.A. sports event is sometimes just as valuable as the tickets. The second item was a note imprinted “From the Desk of Harvey Davenport,” which said, “You were right, buddy.” It was the third item that won the “most favored item” prize. It was a check made out to the firm of Lennox and Carmichael for $10,000!

  It didn’t require too much brainwork to realize that Pete had tipped Davenport off to the individual responsible for doping the bananas. What else could be worth ten thousand bucks?

  “So that’s what you were whispering to Davenport at the end of the game,” I remarked.

  “Yeah.” Pete was being unusually laconic, even for Pete.

  “Well, I give up,” I said. “I’ve thought about it quite a bit. So, apparently, has everyone else in L.A. Who did it?”

  “I couldn’t be sure, but I thought that Hugh Dryden was the most likely candidate.”

  “Dryden? But he wouldn’t even have that job if it weren’t for Stankowitz! As a matter of fact, he’s ridden on Stankowitz’s back ever since he was his high school coach!”

  “Exactly my point.” Pete poured himself a cup of coffee. “Do you remember when we first noticed that Stankowitz wasn’t on the field? What did Dryden do?”

  I thought back. “Just what he was doing before. He checked over the game plan with Woodhouse.”

  “That’s what I didn’t understand. Here’s Stankowitz, Dryden’s meal ticket, not showing up for the most important game in his career. Wouldn’t you think that Dryden would have been nearly beside himself with worry? Instead, he carried on as if it was business as usual.”

  I looked at him with surprise, liberally mixed with admiration. “Pete! Don’t tell me you’re becoming a student of human nature!”

  He almost blushed. “Well, I couldn’t be sure. That’s why I told Davenport that he should get a team of men to watch Dryden. There was a chance that he’d give himself away, either by trying to cover up something or by collecting his payoff.”

  I took the check and went to the bank, making sure that it got there before it closed. Banks close early on New Year’s Eve, and I always like checks to clear as soon as possible—even if they’re made out by people as wealthy as Harvey Davenport. My conscience was twitching, and I never like that. On the other hand, Pete and I are a partnership, and sometimes I do much more than 50% of the work. It was a problem, but I finally solved it to my satisfaction. I deposited $9,450 in our account, and took the remaining $550 to give to Pete to cover his associated gambling losses, deciding that, in this case, it was probably a justifiable business expense. After all, I reasoned, if Pete hadn’t programmed the Traveling Salesman Problem to get us to arrive at Harvey Davenport’s party just when Davenport had a couple of extra tickets, we would have been out ten thousand bucks.

  I hadn’t really felt like going out this New Year’s Eve. Sometimes I get pangs of loneliness around holidays, and I was feeling a little blue. Pete and Julie Rydecki had recently become an item and had made reservations at one of the trendier nightclub-restaurants, but Julie had come down with the latest incarnation of Asian flu and had canceled.

  Pangs of loneliness are often exacerbated by the wrong company, and I just didn’t want to go to a party with lots of couples. Pete didn’t either, so we decided to throw a couple of steaks on the indoor grill and simply toast the New Year. There’s always a football game on New Year’s Eve, and as I had nothing to do and didn’t want to go back to an empty guesthouse, I watched it with Pete. The episode of The Proud and the Passionate in which Julie had a bit part was due to rerun a little later, and Pete had obviously planned to watch it with Julie until she came down with Asian flu. You may recall that not only hadn’t we seen that episode the first time it showed, but we hadn’t recorded it either, so this gave Pete another chance. After the episode, he got on the phone with Julie about half an hour before midnight, and it was my cue to exit. I wished him Happy New Year, and headed back to the guesthouse.

  I switched on the light.

  “Happy New Year, Freddy.”

  I had a hard time believing my ears, but my eyes were there to visually confirm that it was Lisa! And not merely occupying space, but wearing the off-one-shoulder black dress that always took my breath away every time I saw her in it.

  “It’s certainly starting out that way.” I did have enough breath for that remark. Questions were vying with each other for voice time, and the first two to emerge were “How did you get in? Did I leave the door unlocked?”

  “No, Pete sent me a key.”

  My jaw dropped. “He what?”

  “He sent me a key. I called him up and told him that our firm was opening an office in Los Angeles, and I had been asked to head it. That was the job opportunity I told you about that I didn’t want to jinx. I said that I wanted to surprise you for New Year’s—that is, if you wanted to be surprised. Pete told me that if there had been a line posted in Vegas on whether or not you wanted to be surprised, they would have taken it off the board. I took that as a yes.”

  I didn’t know whether Pete’s assessment of my feelings on this matter had taken place before or after he told Harvey Davenport that he should keep a close eye on Hugh Dryden, but maybe Pete really was becoming a student of human nature.

  Lisa paused for a moment. “It’s a little chilly in here.” Possibly her choice of outfit had something to do with that statement. “Maybe you could light the fire.”

  I went over to the old-fashioned fireplace. Lisa had arranged the logs and kindling carefully above paper to be used as tinder. Detectives are trained to be observant, and I saw that Lisa hadn’t used the available old newspaper for that purpose, so I looked a little more closely at the paper. It was our separation agreement!

  “Good choice of tinder,” I said approvingly.

  I made sure that the flue was open, as this was clearly one of those moments that would have been ruined by the buzzing of a smoke alarm, and lit one of the long matches. We watched the paper burn, and then the kindling. Just as the logs were beginning to catch fire, the clock struck midnight. I could hear sirens, horns honking, and—as happens every New Year’s, celebratory gunshots from the general direction of Venice, which borders Santa Monica.

  “Happy New Year, Lisa.”

  We kissed—and not just a Happy New Year kiss. When we finally broke apart, Lisa asked, “Do you think we should go over and wish Pete a Happy New Year?”

  “It can wait until morning,” I replied, as I put my arms around her.

  APPENDIXES

  CONTINUING THE INVESTIGATIONS

  APPENDIX 1

  MATHEMATICAL LOGIC IN “A CHANGE OF SCENE”

  Before you dive into this section, let me repeat something I said earlier—you don’t have to read this! I think a fair amount of learning takes place by erosion—if you’re simply exposed to something often enough, it will sink in. Maybe not deeply, but enough to give you the idea. Hang around with musicians, you get some idea of what goes into music—maybe not enough to make you a musician yourself, but you’ll be a lot more knowledgeable about it than if you spent no time on it at all.

  However, I hope that you’ll try reading some of these sections. They’re not deep, and if you get fed up, you can always go on to the next story.

  Sherlock Holmes was fond of telling Watson that when you eliminate the impossible, whatever remains, no matter how improbable, must be the truth. It’s certainly a simple, insightful, and elegantly phrased remark. However, it does not come as a stunning surprise because most of us are already aware of the inherent logic behind Holmes’s statement.

  It is perhaps fitting that Sherlock Holmes, for whom logic was a sine qua
non, was an Englishman, for it was another Englishman, George Boole, who was most responsible for the invention of symbolic logic.

  Before George Boole, mathematicians concerned themselves with mathematical objects such as numbers and geometric figures. A goal of mathematics then, as now, was to prove theorems—about numbers, geometric figures, and such. To George Boole goes the honor of being the first extensive investigator of the nature of proof (not to slight the Greek philosophers, who made the initial contributions in this area).

  One of the reasons that mathematics is so successful is its relentless focus on concepts whose definitions are unambiguous. Boole focused his attention on statements or sentences that were either unambiguously true or unambiguously false. Such statements are called propositions. Throughout appendix 1, the letters P, Q, and R are used to denote propositions, and the letters T and F are used as abbreviations for “true” and “false,” respectively.

  To each proposition there is an opposite, which is called NOT P. A proposition can be negated simply by sticking the phrase “It is false that …” in front of the proposition. For instance, if P is the proposition “Today is Thursday,” the opposite of P (sometimes called the negation of P) is the proposition “It is false that today is Thursday.”

  From simple propositions, more complicated ones can be built up through the use of the logical terms OR and AND, which behave pretty much the way they do in ordinary English. The proposition P AND Q, for instance, is true only when both P and Q are true—just as you would expect. If someone said to you, “The New England Patriots won the Super Bowl in 2015, and Austin is the capital of Texas,” you’d undoubtedly agree that the statement was true, although maybe you’d need to do a little Googling first. However, if they said, “The New England Patriots won the Super Bowl in 2015, and 2 + 2 = 5,” that would elicit a “whoa, there” on your part.

  The term OR is a little more subtle because we use the word “or” in English in two different ways. The exclusive “or” is used to preclude one of the two possibilities, as in “Did you register as a Democrat or as something else?” You can’t do both—at least, not legally. However, the inclusive “or” allows both possibilities to be selected. When the waiter at a restaurant asks you if you would like coffee or dessert, he obviously won’t be offended if you choose to have both—especially as the size of his tip will probably increase. At some time in the past, mathematicians decided to use the inclusive “or,” just as they decided to use the symbol + for addition rather than something else—and so we’ll go with that.

  This information can be quickly summarized in tabular form. The layout that follows is known as a truth table.

  With the four rows, we have covered all the possible true-false combinations for the two propositions P and Q, and the other columns give the truth values of the proposition at the top of the column for the truth values of P and Q in the same row.

  Here’s a very informative, very short truth table.

  In words, for any proposition P, the proposition P OR NOT P is always true, and the proposition P AND NOT P is always false. Propositions that are always true are known as tautologies.

  Some truth tables are important but not especially interesting. It is easy to show that P OR Q and Q OR P have the same truth table. This isn’t surprising, as if your waiter asks, “Will you have dessert or coffee?” it’s the same question as if he asked, “Will you have coffee or dessert?” Similarly, P AND Q and Q AND P have the same truth table.

  It is possible to use parentheses to construct ever more complicated propositions, much as parentheses are used in arithmetic and algebra for exactly the same purpose. If P, Q, and R are propositions, we can construct the compound proposition P OR (Q AND R) by first constructing the proposition Q AND R and then taking that proposition and OR-ing (as the computer folk are fond of saying) the proposition P with the proposition Q AND R.

  We can compute the truth value of a complicated proposition from the truth values of its components simply by stripping away levels of parentheses. Just as we compute the numerical expression (2 + 3) × (3 × (4 + 7)) by working from the inside out, we can do the same thing with complex propositions.

  Arithmetically, (2 + 3) × (3 × (4 + 7)) = 5 × (3 × 11) = 5 × 33 = 165. To see how this works in symbolic logic, we’ll assume that P is true, Q is false, and R is false. To compute the truth value of the following logical expression, (P OR NOT Q) AND (NOT P OR R), we just replace each proposition by T or F as we compute it.

  1) (P OR NOT Q) AND (NOT P OR R)

  2) (T OR NOT F) AND (NOT T OR F)

  3) (T OR T) AND (F OR F)

  4) T AND F

  5) F

  Now let’s return to Sherlock Holmes. How can we analyze his remark that, when you have eliminated the impossible, whatever remains, however improbable, must be true?

  Let’s suppose that P and Q are propositions such that P OR Q is true. Suppose further that Q is false. How can we conclude that P must be true?

  One look at the truth table for P OR Q should make it fairly obvious.

  Since P OR Q is true, line (4) is eliminated. Since Q is false, lines (1) and (3) are likewise out. No matter how improbable P may be, it must be true, as line (2) is the only one remaining, and P is true in line (2).

  Now things get a little complicated. Boole decided to assess the validity of the argument IF P THEN Q on the basis of the true-false values of the propositions P and Q. What Boole decided was that the important thing was to make sure that any argument which started with a true premise (the premise is the P in IF P THEN Q) and ended with a false conclusion (the conclusion is the Q in IF P THEN Q) would be labeled as false. After all, if you start with the truth and reach a false conclusion, your argument must be fallacious. To single out these fallacious arguments, Boole made all other IF P THEN Q statements true, by fiat.

  This resulted in the following truth table for IF P THEN Q.

  P

  Q

  IF P THEN Q

  T

  T

  T

  T

  F

  F

  F

  T

  T

  F

  F

  T

  Let’s look at the compound proposition IF ((P OR Q) AND NOT Q) THEN P. So that everything will fit on one line, let R denote the proposition (P OR Q) AND NOT Q.

  No matter what the truth values of P and Q, the proposition

  IF ((P OR Q) AND NOT Q) THEN P

  is always true!

  Admittedly, when Sherlock Holmes used it, he assumed implicitly that either P or Q is true. Nonetheless, no matter what the truth values of P and Q, IF ((P OR Q) AND NOT Q) THEN P must be true.

  When two propositions have the same truth table, they are said to be logically equivalent, and we often use those equivalences in everyday speech. When the waiter asks you if you would like coffee or dessert and you tell him, “No,” he knows that you do not want coffee and you do not want dessert. You—and the waiter—have used the fact that NOT (P OR Q) is logically equivalent to (NOT P) AND (NOT Q), and Pete used it in the story to conclude that the contact was not going to meet Hazlitt and was also not going to meet Burns.

  Boolean logic, as this branch of mathematics is known, has gone far beyond what Boole could ever have imagined. Not only is your computer constructed on its principles, but also every time you do an advanced search with a search engine, you are using Boolean logic as well.

  APPENDIX 2

  PERCENTAGES IN “THE CASE OF THE VANISHING GREENBACKS”

  As Pete observes in the story, percentages are a source of substantial confusion. Most of this confusion comes from a mistaken belief that percentages work the same way as numbers, with a gain of 20% compensating for a loss of 20%. As we saw in the story, a gain of 20% does not compensate for a loss of 20% because the 20% gain is not figured using the same amount as the 20% loss.

  A knowledge of basic algebra can be quite helpful in eliminating much of the confusi
on surrounding percentages. There’s a little algebra in the material that follows but hopefully not enough to cause you sleepless nights.

  Innumeracy, as Pete points out in the story, is the arithmetic equivalent of illiteracy. Those who have succumbed to illiteracy, however, realize that they cannot read. They know that it can profoundly affect their lives and often take steps to remedy this problem.

  Innumeracy is much more insidious than illiteracy. The victims often do not realize that they are innumerate. Illiteracy is condemned, but innumeracy is not regarded in the same light. There are those who feel that such attributes as artistic creativity go hand in hand with innumeracy—indeed, there are even some who proudly flaunt their innumeracy.

  This is a great pity because, like any disease, the ripple effects of innumeracy spread throughout our society. It is probably not an exaggeration to say that elimination of innumeracy would save our society tens, and perhaps hundreds, of billions of dollars annually.

  Any study of innumeracy would undoubtedly find that confusion concerning percentages is a contributing factor. The term “per cent” is an abbreviation of the Latin phrase “per centum,” which means “each hundred.” Thus 3% means 3 for each hundred; 3% of 100 is 3, and 3% of 400 is 12.

  COMPUTING PERCENTAGES

  To find a percentage of a given number, you just need to multiply the number by the percentage, and divide by 100.

  Example 1: To find 7½% of $350, multiply 7½ by $350, obtaining $2,625, and then divide by 100 to get $26.25.

  Computing percentages is straightforward, and most people do not have too much difficulty doing so. Finding the cost of an item on which the sales tax is known is a little more difficult and requires setting up and solving a simple equation.

  Suppose we want to find the purchase price of an item on which a 7% sales tax came to $11.20. Let P denote the purchase price. Then 7% of P is 7 P/100 = 0.07 P. Therefore,