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Logic puzzles can teach reasoning in a way so fun most people don't realize its work.
- Logician Raymond Smullyan devised tons of logic puzzles, but one was declared by another philosopher to be the most difficult of all.
- 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.
Raymond Smullyan, master of logic puzzles, magic, and looking like Gandalf.<iframe width="730" height="430" src="https://www.youtube.com/embed/E27v83WWiGo" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><p> This puzzle was written by the brilliant logician <a href="https://mathshistory.st-andrews.ac.uk/Biographies/Smullyan/" target="_blank" rel="noopener noreferrer">Raymond Smullyan</a>. 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. </p><p> 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 <a href="https://plato.stanford.edu/entries/goedel-incompleteness/" target="_blank" rel="noopener noreferrer">Gödel's incompleteness theorems</a>, in a fun, non-technical way. </p><p> Skilled in close up magic, he once worked as a professional <a href="https://www.nytimes.com/2017/02/11/us/raymond-smullyan-dead-puzzle-creator.html" target="_blank" rel="noopener noreferrer">magician</a>. He was also an accomplished <a href="https://www.youtube.com/watch?v=yamGR6XpwZw" target="_blank" rel="noopener noreferrer">pianist</a> and an amateur astronomer who built his telescope. Besides his interest in logic, he also admired Taoist philosophy and published a book on it for a general audience. <br> </p><p> He also found the time to appear on <a href="https://www.youtube.com/watch?v=E27v83WWiGo" target="_blank" rel="noopener noreferrer">Johnny Carson</a>. In this appearance, as in many of his books, he argues that people who like his puzzles claim to dislike math only because they don't realize that they are one and the same.</p><p> But now, on the on to the main event!</p>
The Hardest Logic Puzzle Ever OR: The Three Gods Problem<iframe width="730" height="430" src="https://www.youtube.com/embed/UyOGZk7WbIk" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><p> One of the more popular wordings of the problem is:<br> <br> "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 <em>yes</em> and <em>no</em> are <em>da</em> and <em>ja</em>, in some order. You do not know which word means which."<br> <br> 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 lair, not merely between answering "da" and "ja." <br> <br> Give yourself a minute to ponder this; we'll look at a few answers below. Ready? Okay. <strong><br> <br> </strong>George Boolos' <a href="https://www.pdcnet.org/8525737F00588A37/file/31B21D0580E8B125852577CA0060ABC9/$FILE/harvardreview_1996_0006_0001_0060_0063.pdf" target="_blank" rel="noopener noreferrer">solution</a> focuses on finding either True or False through complex questions. </p><p> 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. </p><p> 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) </p><p> 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. </p><p> 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: <br> </p><p> "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."<br> <br> 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. </p><p> 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. </p><p> 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. </p><p> 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. </p><p> 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 <a href="https://www.nytimes.com/interactive/2017/02/11/obituaries/smullyan-logic-puzzles.html" target="_blank" rel="noopener noreferrer">interactive</a> for those of you in a rush. </p>
Another amazing tardigrade survival skill is discovered.
- Apparently, some water bears can even beat extreme UV light.
- It may be an adaptation to the summer heat in India.
- Special under-skin pigments neutralize harmful rays.
Stressor testing<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNDU1MzIzMS9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYyMjc2MDc4Mn0.5R6DAfzsq29zvETCEH1sR9rprcnJv_L0KyUW2qedslE/img.jpg?width=980" id="c6b71" class="rm-shortcode" data-rm-shortcode-id="aff70e67a56cdbbdc46b80e72d121ddb" data-rm-shortcode-name="rebelmouse-image" />
3D illustration of a tardigrade
Credit: Dotted Yeti/Shutterstock<p>It seems at times like scientists enjoy playing the "let's see if <em>this</em> kills them" game with tardigrades, a game that humans usually lose. After searching the campus of the Indian Institute of Science, researchers gathered some water bears and brought them back to the lab to see what they could handle.</p><p>The scientists found that after they exposed <a href="http://cshprotocols.cshlp.org/content/2018/11/pdb.emo102301.full" target="_blank"><em>Hypsibius exemplaris</em></a> tardigrades to very high doses — 1 kilojoule (kJ) per square meter — of UV light for about 15 minutes, they would in fact die over the next 24 hours. However, when they aimed the same blasts at the reddish-brown tardigrades…nothing. The humans even quadrupled the UV intensity and, nope, they tracked the water bears for 30 days, and a majority of them, 60%, were still fine.</p><p>As is often the case with tardigrades, the question was how?</p>
Turning deadly light blue<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNDU1MzIwMy9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYxMTM1NTE2N30.n8FiCLgp5aTqmYby2bjpeu9QJRTV7KzaB9tmTHBzWtk/img.jpg?width=980" id="5d4cc" class="rm-shortcode" data-rm-shortcode-id="7aa8735a958123bcfb269920eb4d2aed" data-rm-shortcode-name="rebelmouse-image" />
Tardigrade's normal appearance (left), and under inverted fluorescence (right)
Credit: Suma et al., Biology Letters (2020)<p>When the researchers examined the tardigrades under an inverted fluorescence microscope they found that when they were exposed to UV light, they became blue. The researchers' hypothesis is that these tardigrades carry fluorescent pigments beneath their skin that they deploy as necessary to transform UV light into simple benign, blue light. It may be that this ability has emerged as an evolutionary response to southern tropical India's often-extreme heat. The study says that typical summer-day UV levels in this region are about 4kJ per square meter.</p><p>Of the 40% of the reddish-brown tardigrades that had died before 30 days — mostly after about 20 days — the scientists concluded they had less pigment with which to neutralize UV light.</p><p>When the scientists extracted the pigment from the UV champions and coated some <em>Hypsibius exemplaris</em> tardigrades with the stuff, their resistance to UV exposure was also enhanced, boosting their survival rate to almost twice that of their uncoated brethren.</p><p>"Autofluorescence" has been found in other animals — parrots, scorpions, chameleons, and frogs, among others — so it's not completely unheard-of. Still, surprise, tardigrades seem to be putting it to unusual use by employing it for UV protection. In parrots, for example, autofluorescence is hypothesized to be involved in tweaking coloration during mating rituals. </p>
Rare structures and artifacts of the Viking religion practiced centuries prior to Christianity's introduction has been uncovered by archeologists in Norway, including a "godhouse."
- A 1,200-year-old temple to the Old Norse gods, like Thor, Odin, and Freyr, have been unearthed in Norway by a team of archaeologists.
- It was likely used for worship and sacrifices to gods during the midsummer and midwinter solstices, and other fertility festivals.
- Icelanders are officially practicing the Old Norse pagan religions again; the first temple to the Norse gods in 1000 years is currently being constructed in the City of Reykjavík.
The god house<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNDU1MjM3MC9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTY1ODkzNzY0MX0.pmfs1whuVE2pdAlBLV64Zw0T2FSLPFpm63j48AlyUz4/img.jpg?width=1245&coordinates=0%2C0%2C0%2C0&height=700" id="531ba" class="rm-shortcode" data-rm-shortcode-id="693df469340e1552b5babcf965f8df7e" data-rm-shortcode-name="rebelmouse-image" />
Aerial view of the "godhouse."
Photo courtesy of the University Museum of Bergen<p>The building remains were unearthed by <a href="https://www.universitetsmuseet.no/nb/artikkel/230/enestaende-funn-av-hedensk-gudehov-fra-yngre-jernalder" target="_blank">archaeologists from the University Museum of Bergen</a> in September at the seaside village of Ose located in western Norway ahead of preparations for a new housing development project. Based on the placement of post holes and other artifacts, the team was able to determine the structure of the godhouse and how it was used.</p> <p>The large wooden building was about 45 feet long, 26 feet wide, and 40 feet high, and is thought by archeologists to date from the end of the eighth century. It was likely used for worship and sacrifices to gods during the midsummer and midwinter solstices, which would have been highly revered cosmological events for agrarian societies like the Old Norse.</p> <p> The building's layout is almost identical to late Iron Age godhouses found at Uppåkra in southern Sweden and Tissø in Denmark, but this is the first temple of its kind found in Norway according to archaeologist and architect <a href="https://www.researchgate.net/profile/Soren_Diinhoff" target="_blank" rel="noopener noreferrer">Søren Diinhoff</a> who led the project. </p> <p>"We have discovered the most perfectly shaped godhouse of all the finds so far — I know of no other Scandinavian buildings in which the house construction is as clear as it is here," <a href="https://www.researchgate.net/profile/Soren_Diinhoff" target="_blank" rel="noopener noreferrer">Diinhoff</a>, told <a href="https://www.syfy.com/syfywire/1200-year-old-viking-temple-found-in-norway" target="_blank" rel="noopener noreferrer">Syfy Wire</a>'s Elizabeth Rayne. "I think our building is central to document and verify this very special architecture."</p> <p>Diinhoff told <em>Live Science</em> that god houses at Ose followed the architectural blueprint of Christian basilicas that travelers would have come across in southern regions. Because of this, Old Norse religious temples of this time are characterized by a high tower looming above a pitched roof, similar to early Christian churches. At the site were also a number of cooking pits for preparing religious feats, and a collection of bones — the remains of animal sacrifices.</p><p>Their excavations also revealed traces of early agricultural settlements dating to between 2,000 and 2,500 years ago, including the remains of two <a href="https://www.livescience.com/oldest-viking-settlement-discovered.html" target="_blank" rel="noopener noreferrer">longhouses</a> — large wooden halls typically covered with turf and thatch and used as communal habitations. According to Diinhoff, they would have each been the center of a small farm for a family and their animals. </p>
Old Norse religion<p>Later in the sixth century is when the Norse began to construct large "god houses." These were complex outdoor worship sites dedicated to deities of the Norse pantheon including the fertility god Freyer, the war god Odin, and the storm god Thor. This suggests the worship was more than a small cult or folk practice. Rather, it likely had something to do with elite classes wanting to put on an ideological spectacle. As high-status families began to take control of the earlier religious cults, Norse religious worship became more organized.</p><p>The temple at Ose was likely used for celebrations and sacrifices to gods during the midsummer and midwinter solstices (the shortest and longest days of the year), which would have been highly revered cosmological events for agrarian societies like the Old Norse. Several years ago, a "phallus" stone was found nearby the excavation site. According to Diinhoff, it was likely a part of Old Norse fertility rituals. </p><p>Festivals in which meat, drinks, and treasures were offered to wooden figurines representing the gods would have also taken place. While the gods consumed the spiritual essence of the food and drink, practitioners were able to enjoy the material of the feast. </p> <p> "You would have a good mood, a lot of eating and a lot of drinking," Diinhoff said to Live Science. "I think they would have had a good time."</p>
A return to pagan practices in Norway?<span style="display:block;position:relative;padding-top:56.25%;" class="rm-shortcode" data-rm-shortcode-id="2fc0c3e3cc8af1ce80c73d8eac464841"><iframe type="lazy-iframe" data-runner-src="https://www.youtube.com/embed/OYAWNj76axM?rel=0" width="100%" height="auto" frameborder="0" scrolling="no" style="position:absolute;top:0;left:0;width:100%;height:100%;"></iframe></span><p>Unfortunately, the party was brought to an end during the 11th century. It was then that Norway's rulers imposed Christianity onto the population and tore down and burned pagan religious buildings and demonized the Norse gods. Worship in Christian churches became the law of the land. There's currently no evidence suggesting that Ose's god house was part of the iconoclastic purge, but Diinhoff and his team would like to find out in further work.</p> <p>Recently, <a href="https://bigthink.com/ideafeed/iceland-to-officially-worship-norse-gods-again" target="_self">the Norse pagan religions have made a comeback</a>. For example, an Icelandic neopagan faith group called the <a href="https://asatru.is/" target="_blank" rel="noopener noreferrer">Ásatrú Association of Iceland</a>, is currently one of the country's fastest growing religions. Over the last decade, it's almost quadrupled its membership going from a (granted, low) base of 1,275 people in 2009 to 4,473 in 2018. The association is constructing the first temple to the Norse gods in 1000 years in the City of Reykjavík. The project began in 2017 and after running into a funding roadblock, it's expected to be completed later this year. </p>
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