The Troublesome Triplets
Detective David Davidson had met some strange suspects in his years on the force, but these three took the booby prize. They were the Taylor triplets: Abner, Benji, and Carlo. Despite being identical triplets, the three young men could scarcely have turned out more differently. Abner was the only one of the three who had gone to college, where he had studied Kantian philosophy. Taking Kant very seriously, Abner refused to lie even if it was to save one of his brothers from prison. Benji, in contrast, was a pathological liar who could not tell the truth even if he wanted to, which he didn't. The third brother, Carlo, was the only relatively normal one: sometimes he told the truth and sometimes he lied.
A teller had identified one of the three brothers as the culprit in a brazen bank robbery. The brother had simply walked into the bank in the middle of the afternoon, handed a threatening note to the teller, and walked out with a bag of full of cash. Of course, the teller had not been able to identify which of the three had actually done the deed. As a result, the detective had the triplets brought in for questioning and placed in separate holding cells.
Davidson wasn't even sure which brother was which, and certainly he couldn't charge all three. In fact, he couldn't charge even one until he identified who was who. So, he questioned them one-by-one.
"Which brother are you?" he asked the man in the first cell.
"Who stole the money?"
Davidson moved on to the second cell. "Which of your brothers is in the first cell?" he asked.
"Who stole the money?"
"One of my brothers."
Finally, Davidson asked the man in the third cell which brother was in the first cell.
"Did you steal the money?"
Davidson sighed as he left the third cell. He now had a confession from the man in the first cell, but perhaps he was lying to protect one of his brothers. Until Davidson knew the identities of the three men he couldn't trust the confession.
Can you help Detective Davidson identify which brother is in each cell?
Extra Credit: Which brother stole the money?
Benji is in cell one, Abner in cell two, and Carlo in cell three.
Explanation: Abner could not be the man in the first cell, since he would not lie and claim to be Carlo. Similarly, he could not be the man in the third cell, because he would not claim falsely that he, Abner, was in cell one. Therefore, Abner is in cell two. This means that Benji is in cell one, since Abner says so, and Carlo must be in cell three.
Extra Credit Solution: Carlo stole the money.
Explanation: Abner truthfully stated that one of his brothers stole the money, so either Benji or Carlo did it. However, Benji lied when he confessed to stealing the money. Note that even though Benji claimed to be Carlo, when asked who had stolen the money he said "I did", not "Carlo did." So, Benji was claiming that he, Benji, stole it, but that was a lie. Therefore, Carlo really did steal the money.
Disclaimer: The puzzle you have just read is fictitious. The names have been changed to protect the innocent.
No Orchids, "Which Half?" & the Pyramid of Propaganda
- From the back cover of No Orchids for Miss Blandish by James Hadley Chase1:
"A brilliant piece of writing!" GEORGE ORWELL
From "Raffles and Miss Blandish" by George Orwell:
Now for a header into the cesspool. No Orchids for Miss Blandish, by James Hadley Chase…is not, as one might expect, the product of an illiterate hack, but a brilliant piece of writing, with hardly a wasted word or a jarring note anywhere. …[I]t takes for granted the most complete corruption and self-seeking as the norm of human behaviour. …[S]uch things as affection, friendship, good nature or even ordinary politeness simply do not enter. … Ultimately only one motive is at work throughout the whole story: the pursuit of power. …[T]he book is not in the ordinary sense pornography. … The thing that the ordinary reader ought to have objected to—almost certainly would have objected to, a few decades earlier—was the equivocal attitude towards crime. It is implied throughout No Orchids that being a criminal is only reprehensible in the sense that it does not pay. Being a policeman pays better, but there is no moral difference, since the police use essentially criminal methods. … In a book like No Orchids one is not, as in the old-style crime story, simply escaping from dull reality into an imaginary world of action. One’s escape is essentially into cruelty and sexual perversion. …. In No Orchids anything is ‘done’ so long as it leads on to power. All the barriers are down, all the motives are out in the open. … Several people, after reading No Orchids, have remarked to me, ‘It’s pure Fascism’. This is a correct description, although the book has not the smallest connexion with politics and very little with social or economic problems. … It is a daydream appropriate to a totalitarian age. In his imagined world of gangsters Chase is presenting, as it were, a distilled version of the modern political scene, in which such things as mass bombing of civilians, the use of hostages, torture to obtain confessions, secret prisons, execution without trial, floggings with rubber truncheons, drownings in cesspools, systematic falsification of records and statistics, treachery, bribery, and quislingism are normal and morally neutral, even admirable when they are done in a large and bold way.2
- "The following story was told by Dr. Earl Peacock when he was chairman of the Department of Surgery at the University of Arizona:"
One day when I was a junior medical student, a very important Boston surgeon visited the school and delivered a great treatise on a large number of patients who had undergone successful operations for vascular reconstruction. At the end of the lecture, a young student at the back of the room timidly asked, "Do you have any controls?" Well, the great surgeon drew himself up to his full height, hit the desk, and said, "Do you mean did I not operate on half of the patients?" The hall grew very quiet then. The voice at the back of the room very hesitantly replied, "Yes, that's what I had in mind." Then the visitor's fist really came down as he thundered, "Of course not. That would have doomed half of them to their death." God, it was quiet then, and one could scarcely hear the small voice ask, "Which half?"3
…[I]f propaganda were truly doomed to a future in which it would always be nothing more than a technologically enhanced amplification of…information developed and disseminated by a central source, the future of democracy would be bleak indeed. However, the technology that has enhanced both the nature and dissemination of propaganda has also put its tools in many more rather than in fewer hands. … The new tools of technology are powerful indeed, but they are available to us all. In [the early 20th-century], the culture of information was a pyramid, with propaganda originating at the top and flowing down in a widening cascade from a single source. At least since the proliferation of the Internet and all that is associated with it beginning in the early 1990s, the culture of information has been collapsing, the pyramid flattening out. In the future, it will increasingly approach the intellectual geometry of a perfect plane…and in a flat world, propaganda…will finally become impossible because all available information will be available to everyone at all times. In this way, technology, not great thinkers and their great ideas, may create the apotheosis of democracy.4
Well, that was way too optimistic. I have to admit that I thought much the same way in 2009, but the pyramid has been rebuilt since then.
- James Hadley Chase, No Orchids for Miss Blandish (Avon Books, 1970).
- George Orwell, "Raffles and Miss Blandish", The Orwell Foundation, 1944. Paragraphing suppressed.
- Leon Gordis, Epidemiology (Second edition, 2000), p. 102.
- Alan Axelrod, Selling the Great War: The Making of American Propaganda (2009), p. 226.
Disclaimer: The opinions expressed in the above quotes represent the views of their original authors and do not necessarily represent the views or opinions of The Fallacy Files. The mere appearance of content on this site does not constitute an endorsement by The Fallacy Files or any of its affiliates or assignees.
I've been banging the drum for over fifteen years in favor of the value of guesstimation to critical thinking. The idea behind guesstimation is to estimate some number quickly and easily based on what you already know without doing any research. Guesstimation is not just blind guessing, but educated guessing, that is, guessing based on what you know.
The goal of guesstimation is not to come up with a precise answer to a question: it's an estimate, after all. If you need an exact answer, then you'll have to research rather than estimate, but often you don't need a precise answer and a "ballpark" estimate will do. So, the goal of guesstimation is to use educated guessing to get an estimate that's "in the ballpark".
What's "in the ballpark"? This depends on what you're estimating and how precise an estimate you need. However, one way of defining the "ballpark" is in terms of orders of magnitude (OOMs), that is, tens, hundreds, thousands, millions, and so on. Unlike most puzzles or math problems, in guesstimation there is no right answer, though there are wrong ones. If a guesstimate is the right OOM, then that may be a good enough answer.
How can you learn to guesstimate? There's no algorithm for guesstimation, so the only way to learn to do it is to see examples of how it's done and try it yourself. That's what this entry is all about: you can try your hand at a guesstimation problem, then compare how you did it to how I did it. I'll provide some hints and suggestions along the way, but the main thing is to practice it yourself. As an added bonus, it's fun!
A guesstimate is not just a guess, or even just an educated guess, it's also an estimate. So, don't just try to immediately guess the answer; instead, use what you know to calculate the answer.
So, let's get started. Here's the question:
Guesstimate It: How many American women are currently of childbearing age?*
Extra Credit: What percentage of the total population of the United States are women of childbearing age?
Use what you know. Do you know the population of the United States? Since we're guesstimating, you don't have to know it exactly. It's approximately a third of a billion, that is, 330 million―this is a good "landmark" number to remember, since it will help you to navigate the statistical landscape. You also already know that about half of all people are female, which is just common knowledge. So, there are about 165 million females in the U.S. currently. This, of course, does not answer the question, but it's a step in that direction.
What are the childbearing years? You probably don't know an exact answer to this question, but you have some idea based on common knowledge and experience. You're only trying to estimate the answer, and precision is neither possible nor necessary. A woman's childbearing years begin at puberty, which is usually at some point in the early teens, and end by menopause, which tends to occur in the forties. So, there are roughly 25 to 35 childbearing years. Let's take 30 years as a compromise.
What is the average life span of an American? It might not occur to you to ask this question, but you'll need the answer in order to calculate the fraction of the female population of childbearing years. You ought to be able to estimate this to within a decade based on common knowledge. A good guess is: at least in the 70s, and possibly up to the 80s. Let's take eighty years as a compromise―this is another good landmark number to remember.
Calculate. Take the three guesses from the three previous hints, and calculate how many American women are currently of childbearing age.
From Hint 1, we have estimated that there are currently 165 million females in the United States. From Hint 2, we estimated thirty childbearing years in a typical woman's life. From Hint 3, we estimated a typical woman's lifespan as eighty years. Thus, her childbearing years are thirty years out of eighty, or 30/80ths―that is―3/8ths, of her life. This means that American women of childbearing age are 3/8ths of 165 million, or approximately 62 million. Given that this is a guesstimate, an answer of between 60 and 65 million would be close enough. How close was your estimate?
Extra Credit Answer: This part is now easy. 62 million is about 19% of 330 million.
You may wonder just how good an estimate your guesstimate is, as well as the one that I gave above. To find out, you'll have to do some research. After making the above Guesstimate, I did some research of my own and here is the result: the March of Dimes organization (MoD), a group dedicated to fighting birth defects and reducing infant mortality, defines the childbearing years as 15-44. So, this is very close to the guesstimate in Hint 2. The MoD also informs us that the number of American women of this age was exactly 64,543,832 as of three years ago. That's 65 million, correcting for over-precision. So, the Guesstimate above was only three million off; that's definitely in the ballpark! Was your guesstimate in the ballpark?
"Population of Women 15-44 years by age: United States, 2020", March of Dimes, 1/2022
* This problem was suggested by one from Saul X. Levmore & Elizabeth Early Cook's Super Strategies for Puzzles and Games (1981), pp. 57-58.
Earlier this month, an article in The New York Post claimed: "The USDA recommends drinking eight to 10 glasses of water per day…"1. An earlier Post article, which is virtually an ad for Evian water, attributed the same recommendation to the same agency2, and perhaps is the source of this month's claim.
I've heard the same advice since I was a child, though not attributed to the United States Department of Agriculture (USDA) as far as I recall. The recommendation was always specifically eight glasses, not nine or ten, and specifically water, not other beverages. Even when I was young this seemed absurd to me: was I supposed to drink eight glasses of water in addition to the glasses of milk, orange juice, RC cola, and Shasta root beer I drank? If I had done so, I would have never left the bathroom.
This entry is not a history of the advice to drink eight glasses of water a day, but I did discover that the recommendation is at least a hundred years old. An article in Everygirl's Magazine of March, 1924 asserts: "Most people interested in experiments of the right way to live say that the body requires about eight glasses of water each day3." So, the advice was already well-established when the article was published, and the USDA was not mentioned.
The traditional recommendation is vague in at least two ways: how much is a "glass", and does the water have to be plain or can it be consumed in other beverages or even food? Drinking glasses range in size from shot glasses, which hold only a fluid ounce or two, to pint beer glasses that hold sixteen ounces. Some recent versions specify eight-ounce glasses, and the recommendation is referred to as the "8×8 rule"4, which amounts to 64 ounces or a half gallon. That's a lot, especially if it's in addition to other beverages consumed in a day.
Eight ounces is a standard cup, so why not express the rule in terms of eight cups a day? Of course, "cup" is ambiguous―is it a coffee cup or a measuring cup?―but the rule could make it clear that it's the standard eight-ounce measurement rather than the vessel from which it is consumed.
Surprisingly, The Post itself reported late last year on a study concluding that the rule was incorrect5. Moreover, the USDA's most recent version of "Dietary Guidelines for Americans" makes no recommendation as to how much water or other fluids Americans should consume6.
The Post's article seems to have been mostly cribbed from an NBC News report7, but the NBC version makes no mention of the alleged USDA recommendation. Instead, it attributes the following precise recommendation to the "National Academies of Medicine": eight eight-ounce cups of "fluid" daily. As far as I can tell, there is no National Academies of Medicine, though there are National Academies, and one of them is the National Academy, singular, of Medicine (NAM). However, the NAM's report on Dietary Reference Intakes lists 3.7 liters of water daily for men and 2.7 for women, which translates to over fifteen eight-ounce cups for men and over eleven for women! However, the report also states:
All sources can contribute to total water needs: beverages (including tea, coffee, juices, sodas, and drinking water) and moisture found in foods. Moisture in food accounts for about 20% of total water intake. Thirst and consumption of beverages at meals are adequate to maintain hydration.8
So, you don't actually have to drink any plain water at all to stay hydrated, since most beverages are 90% water or more, and many foods contain water. Moreover, your body will signal you if it needs water by making you thirsty, so that most people don't need to be sweating about how much water they're consuming.
Despite these facts, the scary headline of the Post article suggests that you're more likely to die if you don't drink enough water:
Drink up: Large study finds that not consuming enough water increases risk of death by 20%1
As is typical of these kind of studies that make their way into the headlines, this is an observational study, as opposed to an experimental one. The study finds "links", "associations", and correlations that "suggest" but don't prove things. Such studies are at best preliminary ones that should lead to experiments, but all too often do not.
As I've mentioned before9, many health and science news articles begin and often end as news releases, and this one is no exception. The author of the press release is careful not to suggest that the study establishes causation:
The findings don’t prove a causal effect, the researchers noted. Randomized, controlled trials are necessary to determine if optimal hydration can promote healthy aging, prevent disease, and lead to a longer life. However, the associations can still inform clinical practice and guide personal health behavior.10
Well, they can, but so can your daily horoscope. It probably won't hurt to drink more fluids―at least if you don't overdo it―but without evidence of causation there's no reason to think it will do any good.
Both the NBC and Post articles include the following quote from the study's lead author: "Emerging evidence from our and other studies indicate [sic] that adding consistent good hydration to healthy lifestyle choices may slow down the aging process." We've seen "emerging science", "emerging research", and "emerging data" before11; now we can add "emerging evidence", which is evidence that has not yet emerged. My advice is to wait until the evidence has fully emerged before worrying about how much water you should drink.
- Jeanette Settembre, "Drink up: Large study finds that not consuming enough water increases risk of death by 20%", The New York Post, 1/2/2023.
- SWNS, "Most adults admit they don’t drink nearly enough water every day", The New York Post, 9/3/2020.
- Dorothy Nye, "Playing the Game of Health", Everygirl's Magazine, 3/1924.
- Hrefna Palsdottir, "Drink 8 Glasses of Water a Day: Fact or Fiction?", Healthline, 10/12/2020.
- Adriana Diaz, "The rule you need eight glasses of water a day is nonsense: study", The New York Post, 11/3/2022.
- "Dietary Guidelines for Americans, 2020-2025", United States Department of Agriculture, accessed: 1/8/2023.
- Aria Bendix, "Poor hydration may be linked to early aging and chronic disease, a 25-year study finds", NBC News, 1/2/2023.
- See: "Chapter 4: Water", in: "Dietary Reference Intakes for Water, Potassium, Sodium, Chloride, and Sulfate", National Academies, accessed: 1/8/2023.
- See: "Do you smoke after sex?", 2/14/2021.
- "Good hydration linked to healthy aging", National Heart, Lung, and Blood Institute, 1/2/2023.
Disclaimer: The information in this entry is not intended or implied to be a substitute for professional medical advice, diagnosis or treatment. All content, including text, graphics, and jokes, is for general information purposes only. Please don't sue me.
Prophecy or Prophesy?
Speaking of prophecy, as I was last year, here's an example from a book I recently read of a common mistake: "…[C]rude historical determinism is mostly a self-fulfilling prophesy…"1. The two words "prophecy" and "prophesy" are obviously related, both having to do with predicting the future, but the first is the noun form and the latter is a verb. "To prophesy" is to predict the future, and the prediction that results is a "prophecy". So, the example sentence should have read: "historical determinism is…a self-fulfilling" prophecy.
I've seen this mistake frequently enough that it was already in my mental spell-checker2, and most of the books on usage that I regularly consult mention it3. For those reasons, it seems to be a common misspelling.
In my experience, the misspelling seems to go mainly in one direction, that is, from "prophecy" to "prophesy". It may be that people are unsure how the former word is spelled, since the "c" is pronounced as an "s", but spelling it as it's pronounced produces a different word.
Since "prophecy" and "prophesy" are both English words, but different parts of speech, you might expect that a spell checker would not detect the substitution of one for the other, but a grammar checker should. I tried the example sentence in several programs and a few did indeed catch the mistake and suggest the correct spelling, but as many others missed it. So, you might want to check your own checker to see whether it will catch this error; if not, you can commit it to the checker in your head.
- Yunte Huang, Charlie Chan: The Untold Story of the Honorable Detective and His Rendezvous With American History (2010), p. 152.
- Another recent example is: W. Joseph Campbell, Lost in a Gallup: Polling Failure in U.S. Presidential Elections (2020), p. 239.
- Here, in alphabetical order by author's last name, are the books:
- Bill Bryson, Bryson's Dictionary of Troublesome Words: A Writer's Guide to Getting It Right (2002)
- Robert J. Gula, Precision: A Reference Handbook for Writers (1980), p. 220
- Porter G. Perrin, Reference Handbook of Grammar & Usage (1972)
- Harry Shaw, Dictionary of Problem Words and Expressions (Revised edition, 1987)
- Bill Walsh, Lapsing Into a Comma: A Curmudgeon's Guide to the Many Things That Can Go Wrong in Print―and How to Avoid Them (2002), p. 195