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Can you solve what an MIT professor once called 'the hardest logic puzzle ever'?
Logic puzzles can teach reasoning in a fun way that doesn't feel like work.

- Logician Raymond Smullyan devised tons of logic puzzles, but one was declared by another philosopher to be the hardest of all time.
- The problem, also known as the Three Gods Problem, is solvable, even if it doesn't seem to be.
- It depends on using complex questions to assure that any answer given is useful.
Despite the general dislike of mathematics that most profess to have, many people enjoy logic puzzles. This is strange, as many logic puzzles are just variations of math problems. Gleefully ignorant of this fact, many mathaphobes will try to solve riddles and puzzles of tremendous difficulty using reasoning tools they fear to employ when the subject is an equation.
Today, we'll look at a puzzle, the polymath who devised it, and why you should consider picking up a book of logical puzzles next time you are at the library.
This puzzle was written by the brilliant logician Raymond Smullyan. Born in New York 101 years ago, Smullyan earned his undergraduate degree at the University of Chicago and his doctorate in mathematics at Princeton, where he also taught for a few years.
An extremely prolific writer, he published several books on logic puzzles for popular consumption and an endless stream of textbooks and essays for an academic audience on logic. His puzzle books are well regarded for introducing people to complex philosophical ideas, such as Gödel's incompleteness theorems, in a fun and non-technical way.
Skilled in close-up magic, Smullyan once worked as a professional magician. He was also an accomplished pianist and an amateur astronomer who built his own telescope. Besides his interest in logic, he also admired Taoist philosophy and published a book on it for a general audience.
He also found the time to appear on Johnny Carson, where, as in many of his books, he argued that people who like his puzzles claim to dislike math only because they don't realize that they are one and the same.
The Three Gods Problem
One of the more popular wordings of the problem, which MIT logic professor George Boolos said was the hardest ever, is:
"Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which."
Boolos adds that you are allowed to ask a particular god more than one question and that Random switches between answering as if they are a truth-teller or a liar, not merely between answering "da" and "ja."
Give yourself a minute to ponder this; we'll look at a few answers below. Ready? Okay.
George Boolos' solution focuses on finding either True or False through complex questions.
In logic, there is a commonly used function often written as "iff," which means "if, and only if." It would be used to say something like "The sky is blue if and only if Des Moines is in Iowa." It is a powerful tool, as it gives a true statement only when both of its components are true or both are false. If one is true and the other is false, you have a false statement.
So, if you make a statement such as "the moon is made of Gorgonzola if, and only if, Rome is in Russia," then you have made a true statement, as both parts of it are false. The statement "The moon has no air if, and only if, Rome is in Italy," is also true, as both parts of it are true. However, "The moon is made of Gorgonzola if, and only if, Albany is the capitol of New York," is false, because one of the parts of that statement is true, and the other part is not (The fact that these items don't rely on each other is immaterial for now).
In this puzzle, iff can be used here to control for the unknown value of "da" and "ja." As the answers we get can be compared with what we know they would be if the parts of our question are all true, all false, or if they differ.
Boolos would have us begin by asking god A, "Does "da" mean yes if and only if you are True if and only if B is Random?" No matter what A says, the answer you get is extremely useful. As he explains:
"If A is True or False and you get the answer da, then as we have seen, B is Random, and therefore C is either True or False; but if A is True or False and you get the answer ja, then B is not Random, therefore B is either True or False… if A is Random and you get the answer da, C is not Random (neither is B, but that's irrelevant), and therefore C is either True or False; and if A is Random...and you get the answer ja, B is not random (neither is C, irrelevantly), and therefore B is either True or False."
No matter which god A is, an answer of "da" assures that C isn't Random, and a response of "ja" means the same for B.
From here, it is a simple matter of asking whichever one you know isn't Random questions to determine if they are telling the truth, and then one on who the last god is. Boolos suggests starting with "Does da mean yes if, and only if, Rome is in Italy?" Since one part of this is accurate, we know that True will say "da," and False will say "ja," if faced with this question.
After that, you can ask the same god something like, "Does da mean yes if, and only if, A is Random?" and know exactly who is who by how they answer and the process of elimination.
If you're confused about how this works, try going over it again slowly. Remember that the essential parts are knowing what the answer will be if two positives or two negatives always come out as a positive and that two of the gods can be relied on to act consistently.
Smullyan wrote several books with other logic puzzles in them. If you liked this one and would like to learn more about the philosophical issues they investigate, or perhaps if you'd like to try a few that are a little easier to solve, you should consider reading them. A few of his puzzles can be found with explanations in this interactive.
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How New York's largest hospital system is predicting COVID-19 spikes
Northwell Health is using insights from website traffic to forecast COVID-19 hospitalizations two weeks in the future.
- The machine-learning algorithm works by analyzing the online behavior of visitors to the Northwell Health website and comparing that data to future COVID-19 hospitalizations.
- The tool, which uses anonymized data, has so far predicted hospitalizations with an accuracy rate of 80 percent.
- Machine-learning tools are helping health-care professionals worldwide better constrain and treat COVID-19.
The value of forecasting
<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNTA0Njk2OC9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYyMzM2NDQzOH0.rid9regiDaKczCCKBsu7wrHkNQ64Vz_XcOEZIzAhzgM/img.jpg?width=980" id="2bb93" class="rm-shortcode" data-rm-shortcode-id="31345afbdf2bd408fd3e9f31520c445a" data-rm-shortcode-name="rebelmouse-image" data-width="1546" data-height="1056" />Northwell emergency departments use the dashboard to monitor in real time.
Credit: Northwell Health
<p>One unique benefit of forecasting COVID-19 hospitalizations is that it allows health systems to better prepare, manage and allocate resources. For example, if the tool forecasted a surge in COVID-19 hospitalizations in two weeks, Northwell Health could begin:</p><ul><li>Making space for an influx of patients</li><li>Moving personal protective equipment to where it's most needed</li><li>Strategically allocating staff during the predicted surge</li><li>Increasing the number of tests offered to asymptomatic patients</li></ul><p>The health-care field is increasingly using machine learning. It's already helping doctors develop <a href="https://care.diabetesjournals.org/content/early/2020/06/09/dc19-1870" target="_blank">personalized care plans for diabetes patients</a>, improving cancer screening techniques, and enabling mental health professionals to better predict which patients are at <a href="https://healthitanalytics.com/news/ehr-data-fuels-accurate-predictive-analytics-for-suicide-risk" target="_blank" rel="noopener noreferrer">elevated risk of suicide</a>, to name a few applications.</p><p>Health systems around the world have already begun exploring how <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7315944/" target="_blank" rel="noopener noreferrer">machine learning can help battle the pandemic</a>, including better COVID-19 screening, diagnosis, contact tracing, and drug and vaccine development.</p><p>Cruzen said these kinds of tools represent a shift in how health systems can tackle a wide variety of problems.</p><p>"Health care has always used the past to predict the future, but not in this mathematical way," Cruzen said. "I think [Northwell Health's new predictive tool] really is a great first example of how we should be attacking a lot of things as we go forward."</p>Making machine-learning tools openly accessible
<p>Northwell Health has made its predictive tool <a href="https://github.com/northwell-health/covid-web-data-predictor" target="_blank">available for free</a> to any health system that wishes to utilize it.</p><p>"COVID is everybody's problem, and I think developing tools that can be used to help others is sort of why people go into health care," Dr. Cruzen said. "It was really consistent with our mission."</p><p>Open collaboration is something the world's governments and health systems should be striving for during the pandemic, said Michael Dowling, Northwell Health's president and CEO.</p><p>"Whenever you develop anything and somebody else gets it, they improve it and they continue to make it better," Dowling said. "As a country, we lack data. I believe very, very strongly that we should have been and should be now working with other countries, including China, including the European Union, including England and others to figure out how to develop a health surveillance system so you can anticipate way in advance when these things are going to occur."</p><p>In all, Northwell Health has treated more than 112,000 COVID patients. During the pandemic, Dowling said he's seen an outpouring of goodwill, collaboration, and sacrifice from the community and the tens of thousands of staff who work across Northwell.</p><p>"COVID has changed our perspective on everything—and not just those of us in health care, because it has disrupted everybody's life," Dowling said. "It has demonstrated the value of community, how we help one another."</p>Listen: Scientists re-create voice of 3,000-year-old Egyptian mummy
Scientists used CT scanning and 3D-printing technology to re-create the voice of Nesyamun, an ancient Egyptian priest.
- Scientists printed a 3D replica of the vocal tract of Nesyamun, an Egyptian priest whose mummified corpse has been on display in the UK for two centuries.
- With the help of an electronic device, the reproduced voice is able to "speak" a vowel noise.
- The team behind the "Voices of the Past" project suggest reproducing ancient voices could make museum experiences more dynamic.
Howard et al.
<p style="margin-left: 20px;">"While this approach has wide implications for heritage management/museum display, its relevance conforms exactly to the ancient Egyptians' fundamental belief that 'to speak the name of the dead is to make them live again'," they wrote in a <a href="https://www.nature.com/articles/s41598-019-56316-y#Fig3" target="_blank">paper</a> published in Nature Scientific Reports. "Given Nesyamun's stated desire to have his voice heard in the afterlife in order to live forever, the fulfilment of his beliefs through the synthesis of his vocal function allows us to make direct contact with ancient Egypt by listening to a sound from a vocal tract that has not been heard for over 3000 years, preserved through mummification and now restored through this new technique."</p>Connecting modern people with history
<p>It's not the first time scientists have "re-created" an ancient human's voice. In 2016, for example, Italian researchers used software to <a href="https://www.smithsonianmag.com/smart-news/hear-recreated-voice-otzi-iceman-180960570/" target="_blank">reconstruct the voice of Ötzi,</a> an iceman who was discovered in 1991 and is thought to have died more than 5,000 years ago. But the "Voices of the Past" project is different, the researchers note, because Nesyamun's mummified corpse is especially well preserved.</p><p style="margin-left: 20px;">"It was particularly suited, given its age and preservation [of its soft tissues], which is unusual," Howard told <em><a href="https://www.livescience.com/amp/ancient-egypt-mummy-voice-reconstructed.html" target="_blank">Live Science</a>.</em></p><p>As to whether Nesyamun's reconstructed voice will ever be able to speak complete sentences, Howard told <em><a href="https://abcnews.go.com/Weird/wireStory/ancient-voice-scientists-recreate-sound-egyptian-mummy-68482015" target="_blank">The Associated Press</a>, </em>that it's "something that is being worked on, so it will be possible one day."</p><p>John Schofield, an archaeologist at the University of York, said that reproducing voices from history can make museum experiences "more multidimensional."</p><p style="margin-left: 20px;">"There is nothing more personal than someone's voice," he told <em>The Associated Press.</em> "So we think that hearing a voice from so long ago will be an unforgettable experience, making heritage places like Karnak, Nesyamun's temple, come alive."</p>Dark matter axions possibly found near Magnificent 7 neutron stars
A new study proposes mysterious axions may be found in X-rays coming from a cluster of neutron stars.
A rendering of the XMM-Newton (X-ray multi-mirror mission) space telescope.
Are Axions Dark Matter?
<span style="display:block;position:relative;padding-top:56.25%;" class="rm-shortcode" data-rm-shortcode-id="5e35ce24a5b17102bfce5ae6aecc7c14"><iframe type="lazy-iframe" data-runner-src="https://www.youtube.com/embed/e7yXqF32Yvw?rel=0" width="100%" height="auto" frameborder="0" scrolling="no" style="position:absolute;top:0;left:0;width:100%;height:100%;"></iframe></span>Put on a happy face? “Deep acting” associated with improved work life
New research suggests you can't fake your emotional state to improve your work life — you have to feel it.
What is deep acting?
<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNTQ1NDk2OS9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYxNTY5MzA0Nn0._s7aP25Es1CInq51pbzGrUj3GtOIRWBHZxCBFnbyXY8/img.jpg?width=1245&coordinates=333%2C-1%2C333%2C-1&height=700" id="ddf09" class="rm-shortcode" data-rm-shortcode-id="9dc42c4d6a8e372ad7b72907b46ecd3f" data-rm-shortcode-name="rebelmouse-image" data-width="1245" data-height="700" />Arlie Russell Hochschild (pictured) laid out the concept of emotional labor in her 1983 book, "The Managed Heart."
Credit: Wikimedia Commons
<p>Deep and surface acting are the principal components of emotional labor, a buzz phrase you have likely seen flitting about the Twittersphere. Today, "<a href="https://www.bbc.co.uk/bbcthree/article/5ea9f140-f722-4214-bb57-8b84f9418a7e" target="_blank">emotional labor</a>" has been adopted by groups as diverse as family counselors, academic feminists, and corporate CEOs, and each has redefined it with a patented spin. But while the phrase has splintered into a smorgasbord of pop-psychological arguments, its initial usage was more specific.</p><p>First coined by sociologist Arlie Russell Hochschild in her 1983 book, "<a href="https://www.ucpress.edu/book/9780520272941/the-managed-heart" target="_blank">The Managed Heart</a>," emotional labor describes the work we do to regulate our emotions on the job. Hochschild's go-to example is the flight attendant, who is tasked with being "nicer than natural" to enhance the customer experience. While at work, flight attendants are expected to smile and be exceedingly helpful even if they are wrestling with personal issues, the passengers are rude, and that one kid just upchucked down the center aisle. Hochschild's counterpart to the flight attendant is the bill collector, who must instead be "nastier than natural."</p><p>Such personas may serve an organization's mission or commercial interests, but if they cause emotional dissonance, they can potentially lead to high emotional costs for the employee—bringing us back to deep and surface acting.</p><p>Deep acting is the process by which people modify their emotions to match their expected role. Deep actors still encounter the negative emotions, but they devise ways to <a href="http://www.selfinjury.bctr.cornell.edu/perch/resources/what-is-emotion-regulationsinfo-brief.pdf" target="_blank">regulate those emotions</a> and return to the desired state. Flight attendants may modify their internal state by talking through harsh emotions (say, with a coworker), focusing on life's benefits (next stop Paris!), physically expressing their desired emotion (smiling and deep breaths), or recontextualizing an inauspicious situation (not the kid's fault he got sick).</p><p>Conversely, surface acting occurs when employees display ersatz emotions to match those expected by their role. These actors are the waiters who smile despite being crushed by the stress of a dinner rush. They are the CEOs who wear a confident swagger despite feelings of inauthenticity. And they are the bouncers who must maintain a steely edge despite humming show tunes in their heart of hearts.</p><p>As we'll see in the research, surface acting can degrade our mental well-being. This deterioration can be especially true of people who must contend with negative emotions or situations inside while displaying an elated mood outside. Hochschild argues such emotional labor can lead to exhaustion and self-estrangement—that is, surface actors erect a bulwark against anger, fear, and stress, but that disconnect estranges them from the emotions that allow them to connect with others and live fulfilling lives.</p>Don't fake it till you make it
<p>Most studies on emotional labor have focused on customer service for the obvious reason that such jobs prescribe emotional states—service with a smile or, if you're in the bouncing business, a scowl. But <a href="https://eller.arizona.edu/people/allison-s-gabriel" target="_blank">Allison Gabriel</a>, associate professor of management and organizations at the University of Arizona's Eller College of Management, wanted to explore how employees used emotional labor strategies in their intra-office interactions and which strategies proved most beneficial.</p><p>"What we wanted to know is whether people choose to engage in emotion regulation when interacting with their co-workers, why they choose to regulate their emotions if there is no formal rule requiring them to do so, and what benefits, if any, they get out of this effort," Gabriel said in <a href="https://www.sciencedaily.com/releases/2020/01/200117162703.htm" target="_blank">a press release</a>.</p><p>Across three studies, she and her colleagues surveyed more than 2,500 full-time employees on their emotional regulation with coworkers. The survey asked participants to agree or disagree with statements such as "I try to experience the emotions that I show to my coworkers" or "I fake a good mood when interacting with my coworkers." Other statements gauged the outcomes of such strategies—for example, "I feel emotionally drained at work." Participants were drawn from industries as varied as education, engineering, and financial services.</p><p>The results, <a href="https://psycnet.apa.org/doiLanding?doi=10.1037%2Fapl0000473" target="_blank" rel="noopener noreferrer">published in the Journal of Applied Psychology</a>, revealed four different emotional strategies. "Deep actors" engaged in high levels of deep acting; "low actors" leaned more heavily on surface acting. Meanwhile, "non-actors" engaged in negligible amounts of emotional labor, while "regulators" switched between both. The survey also revealed two drivers for such strategies: prosocial and impression management motives. The former aimed to cultivate positive relationships, the latter to present a positive front.</p><p>The researchers found deep actors were driven by prosocial motives and enjoyed advantages from their strategy of choice. These actors reported lower levels of fatigue, fewer feelings of inauthenticity, improved coworker trust, and advanced progress toward career goals. </p><p>As Gabriel told <a href="https://www.psypost.org/2021/01/new-psychology-research-suggests-deep-acting-can-reduce-fatigue-and-improve-your-work-life-59081" target="_blank" rel="noopener noreferrer">PsyPost in an interview</a>: "So, it's a win-win-win in terms of feeling good, performing well, and having positive coworker interactions."</p><p>Non-actors did not report the emotional exhaustion of their low-actor peers, but they also didn't enjoy the social gains of the deep actors. Finally, the regulators showed that the flip-flopping between surface and deep acting drained emotional reserves and strained office relationships.</p><p>"I think the 'fake it until you make it' idea suggests a survival tactic at work," Gabriel noted. "Maybe plastering on a smile to simply get out of an interaction is easier in the short run, but long term, it will undermine efforts to improve your health and the relationships you have at work. </p><p>"It all boils down to, 'Let's be nice to each other.' Not only will people feel better, but people's performance and social relationships can also improve."</p>You'll be glad ya' decided to smile
<span style="display:block;position:relative;padding-top:56.25%;" class="rm-shortcode" data-rm-shortcode-id="88a0a6a8d1c1abfcf7b1aca8e71247c6"><iframe type="lazy-iframe" data-runner-src="https://www.youtube.com/embed/QOSgpq9EGSw?rel=0" width="100%" height="auto" frameborder="0" scrolling="no" style="position:absolute;top:0;left:0;width:100%;height:100%;"></iframe></span><p>But as with any research that relies on self-reported data, there are confounders here to untangle. Even during anonymous studies, participants may select socially acceptable answers over honest ones. They may further interpret their goal progress and coworker interactions more favorably than is accurate. And certain work conditions may not produce the same effects, such as toxic work environments or those that require employees to project negative emotions.</p><p>There also remains the question of the causal mechanism. If surface acting—or switching between surface and deep acting—is more mentally taxing than genuinely feeling an emotion, then what physiological process causes this fatigue? <a href="https://www.frontiersin.org/articles/10.3389/fnhum.2019.00151/full" target="_blank">One study published in the <em>Frontiers in Human Neuroscience</em></a><em> </em>measured hemoglobin density in participants' brains using an fNIRS while they expressed emotions facially. The researchers found no significant difference in energy consumed in the prefrontal cortex by those asked to deep act or surface act (though, this study too is limited by a lack of real-life task).<br></p><p>With that said, Gabriel's studies reinforce much of the current research on emotional labor. <a href="https://journals.sagepub.com/doi/abs/10.1177/2041386611417746" target="_blank">A 2011 meta-analysis</a> found that "discordant emotional labor states" (read: surface acting) were associated with harmful effects on well-being and performance. The analysis found no such consequences for deep acting. <a href="https://doi.apa.org/doiLanding?doi=10.1037%2Fa0022876" target="_blank" rel="noopener noreferrer">Another meta-analysis</a> found an association between surface acting and impaired well-being, job attitudes, and performance outcomes. Conversely, deep acting was associated with improved emotional performance.</p><p>So, although there's still much to learn on the emotional labor front, it seems Van Dyke's advice to a Leigh was half correct. We should put on a happy face, but it will <a href="https://bigthink.com/design-for-good/everything-you-should-know-about-happiness-in-one-infographic" target="_self">only help if we can feel it</a>.</p>World's oldest work of art found in a hidden Indonesian valley
Archaeologists discover a cave painting of a wild pig that is now the world's oldest dated work of representational art.
