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January 5th, 2026 (Permalink)
"Never Give a Sucker an Even Break"1
Montgomery "Three-Card Monty" Banks2 was sitting on his usual stool at the bar of his favorite local spot when a stranger sat down beside him. "Howdy," Monty began, "May I offer you a sporting proposition." The bartender, who knew Monty well, rolled his eyes but the stranger didn't notice. He just looked at Monty with raised eyebrows.
"I have right here some loose coins," Monty said while patting his pocket."I expect that you too may have some loose change. I suggest we play a little game. We'll each take a handful of coins out of our pockets and drop them in separate piles on the bar at the same time. Then, we'll count the coins in your pile and those in both piles to see if they're even or odd. If they match, you win and I'll pay for your drink; if they don't match, then you pay for mine."
The stranger frowned and looked confused.
"Looky, it's like this," Monty explained, "if the number of coins in your pile is even and the sum of the numbers of coins in both piles is even, you win and I pay for your drink. But if the sum of both piles is odd, you lose and pay for my drink. Same thing if the sum of your pile is odd: if it matches the sum of both piles, you win; if it don't match, you lose. Get it?"
The stranger nodded slowly.
"Now, I don't know how many coins you got in your pocket, and you don't know how many I got. So, there's no way either of us could know the sum of both, is there? Not only that but you don't have to put all your coins on the counter; you can put out any number you want to. There's no way I could know how many. And the number of coins in your pile is odd or even, and the sum of both piles is odd or even, and the payoff is the same. That's a fair bet, ain't it?"
The stranger nodded again.
At this point, Monty and the stranger played the game as described. I won't reveal what happened until later, but here's the question: Is Monty's game a fair bet for the other player, that is, does he have a 50-50 chance of winning?
Extra Credit: If it's not a fair bet, what is the probability that the other player will win?
Monty doesn't like to lose.
Monty never said he doesn't know the parity3 of the number of coins in his pocket.
No, the bet is not fair.
Denouement: Monty won the bet against the stranger who then paid for his own and Monty's drink. After both had finished their drinks, the latter demanded a rematch, suggesting double or nothing so that he could win back what he'd lost. Monty, of course, accepted.
This time the stranger put only about half the coins he had in his pocket onto the bar since he wondered whether Monty had somehow figured out how much change he had. But Monty won again.
Finally, the stranger, now drunk, suggested a final match. He carefully counted the number of coins in his pocket, then chose whether to put an even or odd number on the counter. He lost again, paid the substantial tab, and reeled out of the bar in a huff. The barkeep just shook his head in disgust.
Explanation: Monty's game is a sucker bet, that is, a bet that appears to be fair but is not. Here's how it works: Monty always carries an odd number of coins in his pocket, so when he plays the game only the other player's pile varies in its parity. You may recall from elementary school that the sum of two integers with the same parity―that is, both even or both odd―is even, whereas the sum of two integers with opposite parity―that is, one even and the other odd―is odd. This is why Monty always wins: since the number of coins in his pile is always odd, the number of coins in the other player's pile will be either even or odd, and here's what happens in both cases:
- Even: The sum of an even number and an odd number is odd, so the sum of the two piles does not match the parity of the other player's pile.
- Odd: The sum of an odd number and an odd number is even, so again the parity of the sum of the two piles does not match that of the other player's pile.
In both cases, Monty wins.
Extra Credit Solution: The probability of the other player winning against Monty is zero.
Disclaimer & Disclosure: The story about Monty and the stranger is fictional. The Fallacy Files is not responsible for any money lost, hospital bills, or legal expenses for anyone who imitates Monty. Do not try this stunt at home. Monty is a professional.
I came across this bet in the short story "The Percentage Player" by Leslie Charteris4.
Notes:
- ↑ This saying may have been coined by Wilson Mizner. "Who's Wilson Mizner?" well you may ask. This is why it's usually attributed to W. C. Fields―who titled one of his movies with it in 1936―in accordance with the principle "famous sayings need famous mouths." See: Ralph Keyes, "Nice Guys Finish Seventh": False Phrases, Spurious Sayings, and Familiar Misquotations (1993), pp. 20f, for the principle, and the same author's The Quote Verifier: Who Said What, Where, and When (2006), pp. 214f, for the saying.
- ↑ If you don't know Monty, he's a sharpster who always speaks the truth and nothing but the truth, but he doesn't always tell the whole truth. Also, he never uses sleight-of-hand or gimmicked coins. Monty doesn't manipulate coins; he manipulates minds. For previous puzzles involving Monty, see:
- Three-Card Monty, 4/19/2018
- The Four Ace Puzzle, 6/24/2018
- A Valentine's Day Puzzle, 2/14/2019
- Three-Dice Monty, 3/12/2019
- For All the Marbles, 2/6/2021
- The Return of Three-Dice Monty, 8/4/2021
- The Return of Three-Card Monty, 9/1/2024
- ↑ "Parity", as applied to integers, refers to whether a number is odd or even. See: E. J. Borowski & J. M. Borwein, The Harper Collins Dictionary of Mathematics (1991).
- ↑ Leslie Charteris, "The Percentage Player", The Saint to the Rescue (1959).
January 3rd, 2026 (Permalink)
How to Lie With Notes 4: Ghost Notes
The previous three entries in this series were introductory ones designed to explain the purpose of scholarly notes and how to read them1. Having covered that, we're now ready to get into the primary purpose of the series, namely, to explain how notes can be misleading.
The first way to mislead with notes is what I call "ghost notes", that is, notes that should be there but aren't. There are two types of ghost notes:
- Non-Existent Notes
The first type of ghost notes are those not found in a work that completely lacks notes. Of course, not all books or essays require notes, but those that are supposed to be scholarly should have either notes or some other apparatus2 for checking the work's claims. Such a lack of notes shows a lack of scholarship that should lead to a general skepticism of the work.
For example, I've previously pointed out the lack of notes of any kind3 in Howard Zinn's popular A People's History of the United States4. As a result, it's difficult to determine where Zinn got his history or what exactly is supposed to support it.
In the book, Zinn claims that over 100,000 people died in the firebombing of Dresden, Germany during World War 25, which is three or four times the actual number killed6. In the previous entry where I discussed this claim, I speculated about a possible source for it, but without notes it's hard to be sure. Zinn's book does have a bibliography divided into sections for each chapter, but the chapter in which the claim about Dresden occurs7 is thirty-five pages long, and there are nearly as many sources listed in the bibliography for that chapter as pages8, which makes it guesswork as to which source supports a specific claim. This is why scholars use numbered notes keyed to particular passages, which make it possible to check the noted claims.
A more recent example of missing notes is the "1619 Project", originally published in The New York Times Magazine9. Peter W. Wood, in a critical response to the project, wrote:
The usual way for disputes about history to be resolved is for historians to present their best arguments, and their sources, in journal articles; each side can then examine the evidence for themselves and hammer out the truth. The 1619 Project evades this kind of transparency. …[T]he project as presented in the magazine contains no footnotes, bibliography, or other scholarly footholds.10
The lack of notes was remedied in the subsequent book version of the "project" which added endnotes11.
- Missing Notes
The second type of ghost note is not found in those works that do have some notes. Sometimes in such works one will find a claim or quote that should be noted but isn't. As an example, a book of popular non-fiction, SHAM12, does contain a few endnotes but they are themselves sham notes: instead of references, they are all parenthetical remarks that could have been included in the text.
As a result of the lack of references―there is no bibiography either―it's almost impossible to check the book's many dubious claims. For instance, the book claims: "…[W]hen researchers at the University of Southern California compared students placed in affect-oriented drug-education programs with students receiving no such education, they found that those enrolled in the 'preventive' program increased their use of tobacco by 86.4 percent, alcohol by 42.4 percent, and marijuana by 74.2 percent.13" I was and remain skeptical about this claim due to both the large size of the supposed increases and the glaring over-precision of the percentages given14: there's no way that any such statistics could be accurate to within a tenth of a percentage point, but with no further information on this supposed study provided, I'm unable to check it. As a result, I'm just as skeptical of the book and its claims as I am of its subject.
If a scholarly work does have citations of sources, but lacks one for some specific claim, that should raise the question: Why is that claim unnoted when others are noted? An obvious explanation would be that there's no source for the claim, or that the source is suspect. Unlike the general skepticism that non-existent notes should induce, the lack of a note for a given claim in a work that has notes should arouse a specific skepticism of the unnoted claim. However, if a work has more than a few such ghost notes, that justifies a more general skepticism about the work.
Sometimes what is not there is as important as what is there. When a nonfiction written work either lacks notes entirely, or is missing a note for an important claim, always ask why. Was the author simply too lazy to add notes? Does the author not know the source for the claim? Is the author trying to hide where it came from? None of these possibilities speak well for the work or its author.
Notes:
- ↑ See:
- Introduction, 10/23/2025
- Latin Abbreviations, 11/13/2025
- Anatomy of a Citation, 12/6/2025
- ↑ An extensive or annotated bibliography may go some way to making up for a lack of notes but not all the way, as we'll see.
- ↑ See the addendum to: Goebbels' Final Big Lie, 4/23/2025.
- ↑ Howard Zinn, A People's History of the United States: 1492-Present (Electronic edition, 2009).
- ↑ Ibid., p. 391.
- ↑ "Bombing of Dresden", Encyclopaedia Britannica, 12/8/2025.
- ↑ Ibid., ch. 16.
- ↑ See table of contents and bibliography: "A People's History of the United States", Howard Zinn, accessed: 1/3/2026.
- ↑ The New York Times Magazine, 8/18/2019.
- ↑ Peter W. Wood, 1620: A Critical Response to the 1619 Project (Electronic edition, 2022), pp. 60f.
- ↑ Nikole Hannah-Jones, et al., The 1619 Project: A New Origin Story (2021).
- ↑ Steve Salerno, SHAM: How the Self-Help Movement Made America Helpless (2005).
- ↑ Ibid., pp. 198f.
- ↑ See: Over-precision.
January 1st, 2026 (Permalink)
Struck by "Lightening"
The difference between the almost right word and the right word is really a large matter―it's the difference between the lightening and the lightning.1
Consider my shock when I recently read the following sentence in a book: "Consider that it is far more likely for someone to be struck twice by lightening if they live on a property that attracts electricity.2" "Lightening" is the present participle of "to lighten", meaning to make lighter either in the sense of less dark or less heavy3. So, "lightening" refers to the process of something becoming lighter.
What would it mean for someone to be "struck" by lightening even once, let alone twice? If you take GLP-1 and lose a lot of weight in a short time, are you struck by lightening? I suppose if you put down a heavy object you've been carrying a long time, you might be struck by a sudden sensation of lightness. But what would that have to do with living "on a property that attracts electricity"?
Obviously, the word intended in the sentence was "lightning", with no "e". "Lightning" refers to the familiar electrical discharge that occurs in the atmosphere during storms and is followed by thunder4. Moreover, the phrase "struck by lightning" is the common way of describing what happens when lightning passes through an object. That lightning is an electrical phenomenon explains why living on a property that attracts electricity would increase the chance of being struck.
Misspelling "lightning" by adding an incorrect "e" is a common enough error that I've seen it more than once before coming upon the example above, so my shock at seeing it was not quite like being struck by lightning. Two of my reference books on common errors in English mention the "lightning" versus "lightening" confusion, which is further evidence that this error is common5.
Notes:
- ↑ This is what Mark Twain should have said; what he actually said was: "…[T]he difference between the almost right word and the right word is really a large matter―'tis the difference between the lightning-bug and the lightning." Emphasis in the original. See: George Bainton, ed., The Art of Authorship: Literary Reminiscences, Methods of Work, and Advice to Young Beginners (1890), pp. 87f.
- ↑ James C. Zimring, Partial Truths: How Fractions Distort Our Thinking (2022), p. 102.
- ↑ "Lightening", Cambridge Dictionary, accessed: 12/31/2025.
- ↑ "Lightning", Cambridge Dictionary, accessed: 12/31/2025.
- ↑ The two are:
- Mignon Fogarty, Grammar Girl's 101 Misused Words You'll Never Confuse Again (2011)
- Harry Shaw, Dictionary of Problem Words and Expressions (Revised edition, 1987)
December 25th, 2025 (Permalink)
A Logicians' Club Christmas*
Those fun-loving logicians of the Logicians' Club celebrate Christmas every year with a holiday party, and what a wild party it is! This year every member wore a hat that was either red or green, which were taken to be Christmas colors.
The president of the club was the first to arrive at the venue chosen for the party. The first thing he did was make sure that the room had no mirrors or other shiny surfaces. Not that a club member would willingly cheat, but that he or she might accidentally catch a glimpse of his or her hat in such a reflective surface.
When the four other members arrived, it was time to begin the festivities. The four members were seated in a circle facing each other, and the president walked behind them, pulling a hat out of an opaque bag and placing it on the head of each member in turn. Of course, the logicians could see the colors of the hats on the heads of the other three members, but they could not see the color of the hats on their own heads.
Once the hats were in place, the president issued the following order to the players: "If you can see a red hat, stand up!" All four members stood.
The president continued: "If you can see a green hat, sit down!" All four members sat down.
Finally, the president ordered: "If you know what color your hat is, stand up!" Again, all four members stood.
How many red hats were the four members wearing?
Are you Logicians' Club material? If you can solve this problem, you are!
Extra Credit: How did the four members know the colors of their own hats?
Keep in mind that all members of the Logicians' Club are perfect logicians, which means that if it is possible for them to deduce something based on the information they have, they will do so. Also, members of the club will never lie while playing a game, unless lying is part of the game; lying is not part of this game.
There were two red hats and two green hats.
Explanation: When all four logicians stood up, this meant that there had to be at least two red hats among the four. Obviously, if there were no red hats, no one would have stood up, and if there were only one red hat then the person wearing the red hat would not have stood up since he or she would have seen only green hats. Therefore, there were at least two red hats.
The same is true when all four sat down, with "green" and "red" switched, so there had to be at least two green hats. Given that there were at least two red hats and at least two green hats, it follows that there were exactly two red and exactly two green hats.
Extra Credit Solution: By the time they all sat down again, all four logicians had figured out the number of red and green hats, as explained above, which would have enabled them to deduce the color of their own hats. Each of them would have seen two hats of one color and one of the other, which would have told him or her that his or her own hat was of the other color.
* ↑ If you haven't had enough Christmas puzzles at the Logicians' Club, see: Christmas at the New Logicians' Club, 12/25/2021.
December 24th, 2025 (Permalink)
An Antidote for Christmas?
Though the word "antidotes" sounds and is spelled similarly to "anecdotes", it's hard to think of two nouns whose meaning is more different. An antidote is a type of medicine that prevents or limits the harm of a poison1, whereas an anecdote is a type of short story, often funny and based on one's own or another's experience2.
A story published exactly a year ago included the following anecdote: "When 'A Christmas Story' premiered in 1983, screenwriter Jean Shepherd pulled antidotes from his own life to create the Christmas comedy…3". It's hard to imagine what it would mean for Shepherd to pull antidotes from his life, unless someone poisoned the egg nog. Is there an antidote for fruitcake? Instead, he took anecdotes from his life to write the screenplay.
As I usually do for these entries on easily confused word pairs, I tried the example sentence in several free online spelling and grammar checkers. One checker did correct "antidotes" to "anecdotes", but most others found no errors, though one found a mistake but wouldn't tell me what it was unless I upgraded to "Premium"―no thank you. The only "mistake" that yet another checker found was the pronoun "his", "correcting" it to "her", presumably because of the spelling of Shepherd's first name―again, no thank you.
I don't know how common the confusion of "antidote" and "anecdote" is, and none of the reference books I usually check discuss it, but I've seen it prior to coming across the above example though I don't recall where. But that's just anecdotal evidence.
Notes:
- ↑ "Antidote", Cambridge Dictionary, accessed: 12/23/2025.
- ↑ "Anecdote", Cambridge Dictionary, accessed: 12/23/2025.
- ↑ Here's the original, uncorrected article, courtesy of the Internet Archive's Wayback Machine: Alexandra Bellusci, "Why 'A Christmas Story' writer Jean Shepherd was kicked off set―and Jack Nicholson lost out on role", The New York Post, 12/24/2024; and here's the current, corrected version: Alexandra Bellusci, "Why 'A Christmas Story' writer Jean Shepherd was kicked off set―and Jack Nicholson lost out on role", The New York Post, 12/24/2024.