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February 14th, 2019 (Permalink)

A Valentine's Day Puzzle

"Three card" Monty1, that mind-manipulating mountebank, is back on the midway. A young couple, arm-in-arm and obviously deeply in love, were walking past his booth. "Step right up, sir, and win a prize for your young lady," he called out to the couple, who approached him.

"I have here a pair of dice," Monty said, holding out his hands, palms up, for the couple to see lying on each palm a plastic cube with numbers on its six faces. "As I'm sure you know, a standard die has the numbers 1 through 6 on its faces, for an average2 value of three-and-a-half points a roll. The dice that I am holding are non-standard, with different numbers on their faces such that none are shared by both dice, making ties impossible when they play against one another. Even though these dice are numbered differently from standard ones, they are in every other respect fair dice; they're not loaded or rigged in any way.

"As you can plainly see," Monty continued, "the die in my left hand is numbered with even numbers: sixes on four of its faces and zeroes3 on two for an average of four points per roll. The right-hand one has odd numbers: ones on four of its faces, but sevens on the remaining two, for an average of only three points a roll.

"To play the game, you will select a die and I will take the other, then we will roll against each other and the one who rolls the highest number of points wins.

"Before you choose," Monty continued, "I offer you an even money bet that my die roll beats yours. That is, I will bet you dollar to dollar that the number on my die will be higher than that on yours: if it is, I win a dollar; if not, you win one. Does that seem fair?"

The young man smiled broadly and said: "Very fair."

Can you help the young couple decide which die to choose?



  1. In case you don't know Monty: he is a trickster, but he always speaks the exact truth. So, what Monty said about the dice is the truth and nothing but. However, this does not mean that it's necessarily the whole truth. Also, while he is a sharpster, Monty prides himself in not doing any sleight-of-hand, which means that he would not switch the dice or engage in any other legerdemain. For previous puzzles involving Monty, see:
  2. Throughout this puzzle, "average" refers to the mean value, that is, the total number of points on the die's faces divided by six.
  3. Zero is not usually referred to as an even number, but it fits the usual definition of an integer evenly divisible by two.

February 13th, 2019 (Permalink)

Rule of Argumentation 31: Focus on claims and arguments!

The way I usually like to put this rule is: "Keep your eye on the ball!". Unfortunately, most people won't know what "the ball" is and others may not understand the sports metaphor.

As I pointed out in the Introduction to this series, the most general rule of relevance in argumentation is simply: Be relevant! However, it doesn't help much to tell someone to be relevant if they don't know what's relevant. In argumentation, what's logically relevant relates to what I'm calling here "the ball".

Imagine that an argument is like a tennis game2. Your goal is to hit the ball over the net, not to hit the other player with it, let alone to hit him with your racket. Another good sports-related way to put this rule is: "Play the ball, not the man!" This means to keep your eye on the ball and don't get distracted by the other player.

What is "the ball" in the analogy of argumentation to a tennis game? It is the topic or subject being debated, that is, the claim or claims that the arguers think they disagree about3. In formal debates, there is usually an explicit proposition or resolution that is the topic of the debate, which is "the ball" to keep your eye on. However, the kind of informal debates that most of us engage in most of the time lack a specific topic, which is a major source of difficulty.

Before you can keep your eye on it, you first have to spot the ball. Whenever an argument begins, you should ask yourself: What are we arguing about? If you're not clear about this―and I suspect that much of the time you won't be―how can you expect the other player to be? Often, the players involved don't even agree on what they're arguing about, which is like trying to play a game of tennis with two balls, with each player trying to hit a different ball. Even if you think you know what it is, you should ask the player on the other side what you disagree about. Many an argument is resolved at this stage when the arguers discover that they really don't disagree. However, if you skip this stage, it's possible that the argument may continue indefinitely, with both arguers arguing past each other. Such arguments are frustrating and can easily lead to bad feelings between the arguers, even though they don't actually disagree!

One underappreciated achievement of argumentation is the discovery that you don't substantively disagree with the other side; rather, you just express the same view in different words. However, the only way you will discover this is if you play the ball―the claim or claims you seem to disagree about―and not the other player. If instead of aiming at the ball, you try to hit the other player, you will only make the disagreement between you worse.

It's hard to resist the temptation to play the man instead of the ball when the other player is trying to play you. You may naturally feel that you have to defend yourself by replying in kind. Unfortunately, if the other player won't argue cooperatively, there may not be much that you can do except to refuse to play with such a person. This is another way in which cooperative argumentation is like playing tennis: it takes two willing partners.

In future entries, we'll look in more detail at ways you can lose sight of the ball and end up playing the man, instead. For now, keep in mind that the goal of this game is not to defeat your opponent, but to use arguments to put claims to the test.

Next Month: Rule 4


  1. Previous entries in this series:
  2. Many games with a ball and two players or teams would work, so feel free to substitute your favorite.
  3. At least for the sake of the argument: some arguers play Devil's advocate.

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January 31st, 2019 (Permalink)

A Numismatic Puzzle

Why are 2001 pennies worth more than $20?


January 30th, 2019 (Permalink)

Recommended Reading

Fake news is the theme this month:

January 26th, 2019 (Permalink)

Rule of Argumentation 21: Be ready to be wrong!

I beseech you, in the bowels of Christ, think it possible you may be mistaken.2

Speaking of being wrong, as I admitted to in the previous entry, the next time you find yourself in an argument, be prepared to admit your mistakes. Put yourself in the place of those you argue with for a moment: how would you feel if they simply couldn't be convinced to change their minds under any circumstances? I refuse to argue with people like that.

It's perhaps an unfortunate, but unavoidable, fact that we talk about argumentation as a form of conflict. In fact, argumentation is a substitute for physical conflict: instead of fighting, we argue. So, it is perhaps natural that we use the language of physical conflict when speaking of arguments. The very word "argue" is ambiguous between a verbal quarrel and an attempt to reason with one another―throughout this series I use the word in the latter sense.

I use the analogy between argumentation and conflict3 as much as anyone, because it's awkward to do otherwise. The metaphor is so embedded in our language and thought about arguments that it's hard to avoid4, but it can be misleading5.

We call those we argue with our "opponents", the arguments that we make are "attacks" or "defenses", and everybody wants to "win" an argument. Many books have been published that promise to teach you "how to win an argument", but none how to lose one6 since nobody wants to "lose". But what does it mean to win or lose?

What if you're wrong? What if you're on the wrong side, taking the wrong position? Do you still want to "win"? Wouldn't it be better if you "lost"? Or, to put it another way, wouldn't you be a winner if the argument led you to switch from a false belief to a true one7?

Instead of thinking of argumentation as a type of conflict8, let's think of it as cooperative. You and your partner in argumentation are working together to test a claim: one of you presents the reasons for the claim, and the other the reasons against it. If the debated claim is to be properly tested, it's important that both of you present the strongest cases that you possibly can, but that doesn't make you enemies. When all is said and done, you evaluate the claim. Sometimes it will turn out that you argued on the side of the claim that survives the debate; sometimes it will turn out that you were on the other side.

If you take this approach to argumentation, you will be less tempted to treat your sparring partner as an enemy to be defeated at all costs. Appeals to force, threats, and personal attacks aimed at your partner will seem as out of place to you as they in fact usually are.

However, the most important advantage is to you, the arguer. An argument need not be a fight that you may lose; rather, it can be an opportunity for you to learn. Anyone who gives you good reasons to change your mind is doing you a favor, not harming you. They are giving you a chance to change your false beliefs into true ones. Will you be ready to grasp that opportunity when it is presented to you?

Next Month: Rule 3


  1. Previous entries in this series:
  2. Oliver Cromwell, "Letter to the General Assembly of the Church of Scotland", 8/3/1650. See: John Bartlett, Bartlett's Familiar Quotations, Justin Kaplan, General Editor (16th Edition, 1992).
  3. A related but less misleading analogy is that between arguments and games: it's less dangerous because not all games are zero-sum, that is, those in which one wins at the expense of the other, losing player. Instead, I suggest thinking of argumentation as a win-win game.
  4. I don't intend to even try to do so except in the remainder of this entry.
  5. For a general discussion of misleading analogies, see the fallacy of weak analogy, from the menu to the left.
  6. Unsurprisingly, Amazon's advanced search facility shows no books with the title "How to Lose an Argument".
  7. This, of course, assumes that you accept the distinction between truth and falsehood. Some of the ancient sophists, and some modern ones as well, rejected that distinction. For such sophists, there is only winning or losing.
  8. Some argumentation, such as political debate, takes place in a public forum with an audience. Such arguments are usually given more for the benefit of the audience than the opposing side, and you can't expect one side to jump up in the middle of the debate and admit error, even if the debater suddenly realized it. Moreover, some formal debates have clear rules about who wins and loses; for instance, a lawyer who convinces a judge or jury wins a lawsuit, and the other side loses. For this reason, the analogy with conflicts such as fights and wars is more appropriate to this type of debate than to the informal ones that most of us participate in.

January 12th, 2019 (Permalink)

Faster Than a Speeding Bullet

Our New Book this month1 reminded me of a, hopefully, interesting example. It occurs in the following passage, taken from a book about the Warren Commission's report on the assassination of President Kennedy, discussing the Zapruder film of that event:

The total elapsed time from the end of [frame] 312 to the end of [frame] 313 has reached the seemingly infinitesimal figure of 1/18th of a second. …[A] bullet traveling approximately 1,979 miles per second, the speed of the fatal bullet fired by Oswald, theoretically could cover almost ninety miles in that 1/18th of a second.2

When I first read this passage, it made no particular impression on me. However, I later returned to the book looking for specific information―which I never found―and re-reading the passage, I was struck by the speed given for the bullet: almost 2,000 miles a second sounds extremely fast. Since there are 3,600 seconds in an hour, this means a speed of over seven million miles an hour! Of course, bullets are fast, but are they that fast? "Magic bullet", indeed!

Unfortunately, when it comes to speed, there is a dearth of landmark numbers to use for comparison. In particular, I had little idea how fast bullets fly. Most of us are familiar with speeds of less that a hundred miles per hour, but beyond that our experience is limited to airplanes. How fast is a commercial jetliner? Surely, faster than a hundred MPH. Faster than 200 MPH? Faster than 500 MPH?3 Faster than a bullet?

Other than the speeds of automobiles, planes, and bullets, there are also natural phenomena whose speeds might provide landmarks, specifically, sound and light. Off hand, I didn't know exactly what the speed of sound is4, though I did know that it is faster than airliners5. Are bullets faster than sound? Of course, bullets aren't faster than light because nothing is6. I also knew that the speed of light is approximately 186,000 miles per second, but there is an enormous gap between the speed of sound and the speed of light: a vast desert devoid of numerical landmarks.

Now, I would have bet that the speed of the fastest bullet ever fired is not even close to the speed of light, and much closer to the speed of sound. However, 2,000 miles per second is only slightly more than 1% of the speed of light. Nonetheless, the more I thought about it, the harder I found it to believe that a bullet flies anywhere near that fast.

The above is a reconstruction of my thought process after I read the passage for the second time, and began to have some skeptical qualms about what I was reading. Given the lack of landmark numbers, I couldn't easily check the passage for plausibility by checking it against landmarks. However, there is one other technique for checking suspicious numbers that I've previously used7: cross-checking.

Cross-checking is only possible when there are two or more numbers to check against each other, but that's the case here: in addition to the alleged speed of the bullet, we also have the claim that it would cover almost 90 miles in 1/18th of a second. In fact, anything going 1,979 miles per second would travel closer to 110 miles in an eighteenth of a second. While this isn't a huge discrepancy, it's an odd one, and gives further reason to doubt the numbers.

At this point, the plausibility check of these numerical claims came to an end with the conclusion that they are implausible, even highly implausible, but not definitely wrong. It may not sound like it, but this is a successful plausibility check, since such checks will seldom establish that the numbers checked are definitely wrong. Plausibility checks are not replacements for actual research, but preliminary steps to see whether additional research is called for. In this case, such research is definitely in order.

So, how fast do bullets fly? Unsurprisingly, this varies considerably depending on the type of gun and ammunition, from a low of around 400 feet per second for black-powder muskets to ten times as fast for modern rifles8.

More specifically, how fast was the bullet fired by Oswald? Approximately 2,000 feet per second9, which suggests that somehow "miles" was substituted for "feet" in the passage in question. Given that there are 5,280 feet in a mile, this means that the claims were off by a factor of over 5,000. So, instead of travelling around a hundred miles in an 18th of a second, the bullet would have travelled only about a hundred feet.

So, what have I learned from this experience and what do I hope that you learn from it? Here are the bullet points:


  1. A New Book for a New Year, 1/5/2019.
  2. Richard Warren Lewis, The Scavengers and Critics of the Warren Report (1967), p. 166.
  3. For the record, the average cruising speed of jet airliners is around 550 MPH. Source: "How Fast Do Commercial Jets Fly? ", Reference, accessed: 1/12/2019.
  4. To provide another landmark, the speed of sound is around 750 MPH, which would mean that the fatal bullet was travelling at Mach 10,000! Another reason for skepticism. Source: Patricia Barnes-Svarney, editorial director, The New York Public Library Science Desk Reference (1995), pp. 2 & 18.
  5. The Concorde supersonic transports used to regularly break the sound barrier while flying over the Atlantic ocean between Europe and North America, but they stopped flying fifteen years ago.
  6. At least, if the theory of relativity is correct.
  7. Sobriety Check, Part 2, 3/3/2015.
  8. Patricia Barnes-Svarney & Thomas E. Svarney, The Handy Forensic Science Answer Book (2019), p. 199.
  9. Colin Evans, A Question of Evidence (2005), p. 149.

January 5th, 2019 (Permalink)

A New Book for a New Year: Is That a Big Number?

Numbers are important: getting them wrong has consequences. Numbers give size and shape to our world in all sorts of ways, and we rely on them to inform the decisions we make. It's easy though, when numbers become too large, to become numbed by sheer scale. This is not a book of mind-boggling number facts, or stupefying statistics. This book is all about finding a path through a wilderness of big numbers that we don't grasp as well as we could.1

This new book by author Andrew C. A. Elliott asks an important question. Having not read the book, I don't know whether Elliott makes the point that, as it stands, that question is unanswerable. Except, I suppose, by: It depends.

No number, taken in isolation, is big or small in itself. I'm not referring to the fact that "big" and "small" are vague concepts, though that's true, too. Rather, I mean that whether a number is big, small, or in-between depends on what it quantifies. Seven may seem like a small number, but not if you're talking about human height in feet. Similarly, a million may seem like a large number, but not if you're dealing with the human population of countries.

Of course, some numbers are bigger than others: a million is definitely a larger number than seven. So, whenever you wonder whether a number is big or small, you should ask: relative to what?

One of the uses of numbers is to compare the sizes of sets of things, but we can't do this well if we don't have a sense of the relative sizes of the numbers. Since most of us have little experience with numbers in the millions, billions, and trillions, we tend to have little sense of how such numbers compare2. In the book, Elliott gives five techniques for putting numbers in context3:

  1. Landmark Numbers: This phrase seems to refer to what I've called "statistical benchmarks", "benchmark figures", or just "benchmarks"4. These are numbers that can be remembered or easily looked up, and which can be used to put other numbers into perspective. For instance, it's useful to know that the current population of the United States is about a third of a billion people5. I now prefer Elliott's terminology, since a landmark helps us to find our way around a landscape; in the same way, a landmark number can help us navigate the numerical landscape.
  2. Visualisation: You can see what this must be.
  3. Divide & Conquer: I'm not sure if this is the technique of dividing a big number in order to reduce it to a smaller number that can be more easily "conquered". For instance, the current U.S. national debt is over $21 trillion6, which is way too big for people to grasp. Is that a lot or what? Of course, it would be a lot of money to have on your credit card, but is it a lot for the country to have on its credit card? How can we tell? If you divide the national debt by the landmark number of the current population5, thus giving the per capita debt, the result is close to $65,000. This is an amount within most people's experience, and thus easier to get a sense of its scale. Is that a big number? Since you can now do so for yourself, I'll let you be the judge. In any case, I like the phrase "divide and conquer" for this technique of cutting a big number down to size so that our understanding can conquer it, but I don't know if that's what Elliott means.
  4. Rates & Ratios: The example of "divide & conquer" that I've just given produces a ratio. In understanding risk and safety, rates are usually more useful than absolute numbers. For instance, far more people die in the shower than do so in skydiving accidents, but this doesn't mean that showering is more dangerous than skydiving since a lot more people shower than jump out of airplanes. To properly compare the safety of these activities, we need to take into account the fact that more people do one than the other. To do so, divide the number dying in each activity by the number of those engaging in it.
  5. Log Scales: These are not scales for weighing logs―at least, I don't think they are. Rather, what Elliott is referring to are logarithmic scales, as opposed to the more familiar linear scales. Logarithmic scales are used for measuring such things as earthquakes and sound: both the Richter7 and decibel scales8 are logarithmic. That said, I don't know why Elliott thinks that log scales are so useful that they rank with the previous four methods; I guess that I'll have to read the book to find out.


  1. Andrew C. A. Elliott, Is That a Big Number? (2018), p. 2.
  2. Douglas Hofstadter, an early writer on innumeracy, referred to this as "number numbness". See Metamagical Themas: Questing for the Essence of Mind and Pattern (1985), Chapter 6.
  3. Elliott, p. 5.
  4. See, for instance: Sobriety Check, 2/24/2015. I took this terminology from Joel Best's book, Stat-Spotting: A Field Guide to Identifying Dubious Data (2008), pp. 7-9.
  5. To be more exact, 324 million, according to Wolfram Alpha; accessed: 1/3/2019.
  6. According to Wolfram Alpha; accessed: 1/3/2019.
  7. Bryan Bunch & Jenny Tesar, The Penguin Desk Encyclopedia of Science and Mathematics (2000), under "Richter scale".
  8. Bunch & Tesar, under "decibel".

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