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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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What is human dignity? Here's a primer, told through 200 years of great essays, lectures, and novels.
- Human dignity means that each of our lives have an unimpeachable value simply because we are human, and therefore we are deserving of a baseline level of respect.
- That baseline requires more than the absence of violence, discrimination, and authoritarianism. It means giving individuals the freedom to pursue their own happiness and purpose.
- We look at incredible writings from the last 200 years that illustrate the push for human dignity in regards to slavery, equality, communism, free speech and education.
The inherent worth of all human beings<p>Human dignity is the inherent worth of each individual human being. Recognizing human dignity means respecting human beings' special value—value that sets us apart from other animals; value that is intrinsic and cannot be lost.</p> <p>Liberalism—the broad political philosophy that organizes society around liberty, justice, and equality—is rooted in the idea of human dignity. Liberalism assumes each of our lives, plans, and preferences have some unimpeachable value, not because of any objective evaluation or contribution to a greater good, but simply because they belong to a human being. We are human, and therefore deserving of a baseline level of respect. </p> <p>Because so many of us take human dignity for granted—just a fact of our humanness—it's usually only when someone's dignity is ignored or violated that we feel compelled to talk about it. </p> <p>But human dignity means more than the absence of violence, discrimination, and authoritarianism. It means giving individuals the freedom to pursue their own happiness and purpose—a freedom that can be hampered by restrictive social institutions or the tyranny of the majority. The liberal ideal of the good society is not just peaceful but also pluralistic: It is a society in which we respect others' right to think and live differently than we do.</p>
From the 19th century to today<p>With <a href="https://books.google.com/ngrams/graph?year_start=1800&year_end=2019&content=human+dignity&corpus=26&smoothing=3&direct_url=t1%3B%2Chuman%20dignity%3B%2Cc0" target="_blank" rel="noopener noreferrer">Google Books Ngram Viewer</a>, we can chart mentions of human dignity from 1800-2019.</p><img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNDg0ODU0My9vcmlnaW4ucG5nIiwiZXhwaXJlc19hdCI6MTY1MTUwMzE4MX0.bu0D_0uQuyNLyJjfRESNhu7twkJ5nxu8pQtfa1w3hZs/img.png?width=980" id="7ef38" class="rm-shortcode" data-rm-shortcode-id="9974c7bef3812fcb36858f325889e3c6" data-rm-shortcode-name="rebelmouse-image" />
American novelist, writer, playwright, poet, essayist and civil rights activist James Baldwin at his home in Saint-Paul-de-Vence, southern France, on November 6, 1979.
Credit: Ralph Gatti/AFP via Getty Images
The future of dignity<p>Around the world, people are still working toward the full and equal recognition of human dignity. Every year, new speeches and writings help us understand what dignity is—not only what it looks like when dignity is violated but also what it looks like when dignity is honored. In his posthumous essay, Congressman Lewis wrote, "When historians pick up their pens to write the story of the 21st century, let them say that it was your generation who laid down the heavy burdens of hate at last and that peace finally triumphed over violence, aggression and war."</p> <p>The more we talk about human dignity, the better we understand it. And the sooner we can make progress toward a shared vision of peace, freedom, and mutual respect for all. </p>
With just a few strategical tweaks, the Nazis could have won one of World War II's most decisive battles.
- The Battle of Britain is widely recognized as one of the most significant battles that occurred during World War II. It marked the first major victory of the Allied forces and shifted the tide of the war.
- Historians, however, have long debated the deciding factor in the British victory and German defeat.
- A new mathematical model took into account numerous alternative tactics that the German's could have made and found that just two tweaks stood between them and victory over Britain.
Two strategic blunders<p>Now, historians and mathematicians from York St. John University have collaborated to produce <a href="http://www-users.york.ac.uk/~nm15/bootstrapBoB%20AAMS.docx" target="_blank">a statistical model (docx download)</a> capable of calculating what the likely outcomes of the Battle of Britain would have been had the circumstances been different. </p><p>Would the German war effort have fared better had they not bombed Britain at all? What if Hitler had begun his bombing campaign earlier, even by just a few weeks? What if they had focused their targets on RAF airfields for the entire course of the battle? Using a statistical technique called weighted bootstrapping, the researchers studied these and other alternatives.</p><p>"The weighted bootstrap technique allowed us to model alternative campaigns in which the Luftwaffe prolongs or contracts the different phases of the battle and varies its targets," said co-author Dr. Jaime Wood in a <a href="https://www.york.ac.uk/news-and-events/news/2020/research/mathematicians-battle-britain-what-if-scenarios/" target="_blank">statement</a>. Based on the different strategic decisions that the German forces could have made, the researchers' model enabled them to predict the likelihood that the events of a given day of fighting would or would not occur.</p><p>"The Luftwaffe would only have been able to make the necessary bases in France available to launch an air attack on Britain in June at the earliest, so our alternative campaign brings forward the air campaign by three weeks," continued Wood. "We tested the impact of this and the other counterfactuals by varying the probabilities with which we choose individual days."</p><p>Ultimately, two strategic tweaks shifted the odds significantly towards the Germans' favor. Had the German forces started their campaign earlier in the year and had they consistently targeted RAF airfields, an Allied victory would have been extremely unlikely.</p><p>Say the odds of a British victory in the real-world Battle of Britain stood at 50-50 (there's no real way of knowing what the actual odds are, so we'll just have to select an arbitrary figure). If this were the case, changing the start date of the campaign and focusing only on airfields would have reduced British chances at victory to just 10 percent. Even if a British victory stood at 98 percent, these changes would have cut them down to just 34 percent.</p>
A tool for understanding history<p>This technique, said co-author Niall Mackay, "demonstrates just how finely-balanced the outcomes of some of the biggest moments of history were. Even when we use the actual days' events of the battle, make a small change of timing or emphasis to the arrangement of those days and things might have turned out very differently."</p><p>The researchers also claimed that their technique could be applied to other uncertain historical events. "Weighted bootstrapping can provide a natural and intuitive tool for historians to investigate unrealized possibilities, informing historical controversies and debates," said Mackay.</p><p>Using this technique, researchers can evaluate other what-ifs and gain insight into how differently influential events could have turned out if only the slightest things had changed. For now, at least, we can all be thankful that Hitler underestimated Britain's grit.</p>
We’ve mapped a million previously undiscovered galaxies beyond the Milky Way. Take the virtual tour here.
See the most detailed survey of the southern sky ever carried out using radio waves.
Astronomers have mapped about a million previously undiscovered galaxies beyond the Milky Way, in the most detailed survey of the southern sky ever carried out using radio waves.
A new study shows our planet is much closer to the supermassive black hole at the galaxy's center than previously estimated.
Credit: NAOJ<p><em>Arrows on this map show position and velocity data for the 224 objects utilized to model the Milky Way Galaxy. The solid black lines point to the positions of the spiral arms of the Galaxy. Colors reflect groups of objects that are part of the same arm, while the background is a simulation image.</em></p>
Apple sold its first iPod in 2001, and six years later it introduced the iPhone, which ushered in a new era of personal technology.