Posts Tagged ‘puzzles’
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty”*…
Mark Frauenfelder at Boing Boing with a glorious memory…
This cover from the July 1965 issue of Scientific American illustrates the “Four Bugs Problem” featured in Martin Gardner’s “Mathematical Games” column about op art [see here].
The setup: Four bugs are placed at the corners of a square. They start crawling clockwise (or counterclockwise) at a constant rate, with each bug moving directly toward its neighbor. As the bugs move, they always form the corners of a square that both diminishes in size and rotates. Each bug’s path forms a logarithmic spiral.
Gardner said this can be generalized to any number of bugs starting at the corners of a regular polygon with n sides. In these cases, the bugs will always form the corners of a similar polygon that shrinks and rotates as they move.
Here’s an animated version of the Four Bugs Problem you can try out. If you want to try it with a different number of bugs, go here.
Your correspondent still has his copy of that issue. “The beautiful ‘Four Bugs Problem’” from @Frauenfelder in @BoingBoing.
* Bertrand Russell, A History of Western Philosophy
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As we marvel, we might send carefully-calculated birthday greetings to Ian Stewart; he was born on this date in 1945. As a teenager, he was an avid reader of Gardner’s “Mathematical Games,” from which he developed a love of the subject that led him to become a mathematician who has gone on to make important contributions to the field, especially in catastrophe theory.
But Stewart is more widely known as a popularizer of math– who credits Gardner with modeling the skills needed to be an entertaining communicator. Indeed, from 1991 to 2001 Stewart took over the Scientific American column (which had been renamed “Mathematical Recreations”).
For a list of his (remarkable) books on math and science, see here.
“Chance, too, which seems to rush along with slack reins, is bridled and governed by law”*…
… though that law can sometimes be less than obvious. Erica Klarreich reports on one creative mathematician’s efforts to help us learn…
In late January, Daniel Litt [pictured above] posed an innocent probability puzzle on the social media platform X (formerly known as Twitter) — and set a corner of the Twitterverse on fire.
Imagine, he wrote, that you have an urn filled with 100 balls, some red and some green. You can’t see inside; all you know is that someone determined the number of red balls by picking a number between zero and 100 from a hat. You reach into the urn and pull out a ball. It’s red. If you now pull out a second ball, is it more likely to be red or green (or are the two colors equally likely)?
Of the tens of thousands of people who voted on an answer to Litt’s problem, only about 22% chose correctly. (We’ll reveal the solution below, in case you want to think it over first.) In the months since, Litt, a mathematician at the University of Toronto, has continued to confound Twitter users with a series of probability puzzles about urns and coin tosses.
His posts have prompted lively online discussions among research mathematicians, computer scientists and economists — as well as philosophers, financiers, sports analysts and anonymous fans. Some joked that the puzzles were distracting them from their real work — “actively slowing down economic research,” as one economist put it. Others have posted papers exploring the puzzles’ mathematical ramifications.
Litt’s online project doesn’t just highlight the enduring allure of brainteasers. It also demonstrates the limits of our mathematical intuition, and the counterintuitive nature of probabilistic reasoning. As Litt wrote, there’s “nothing more exhilarating than posing a multiple-choice problem on which 50,000 people do substantially worse than random chance.”…
The answer to this puzzle, other puzzles, and Litt on what makes a great puzzle, and why simple probability questions can be so deceptively difficult: “Perplexing the Web, One Probability Puzzle at a Time,” from @EricaKlarreich in @QuantaMagazine.
Vaguely related (but also very interesting): “The Bookmaker,” via @annfriedman, who observes: “Leif Weatherby and Ben Recht on Nate Silver and the addiction to prediction: ‘Silver insists that viewing all decisions through this lens of gambling is the underappreciated characteristic of Very Successful People,’ they write. ‘But what Silver willfully ignores is that the successful players in this world aren’t the bettors. They are the bookies and casino owners—the house that never loses.'”
* Boethius, The Consolation of Philosophy
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As we contemplate chance, we might send confirmatory birthday greetings to Carl David Anderson; he was born on this date in 1905. An experimental physicist, he shared the 1936 Nobel Prize in Physics for his discovery (that’s to say, confirmation of the existence) of the positron, the first known particle of antimatter… which had been predicted by mathematician and physicist Paul Dirac, whose “Dirac Equation“– in part a product of its author’s application of probability theory– had predicted (among many other features of quantum theory as we know it) the existence of the particle (and antimatter).
“Advantage! What is advantage?”*…
Pradeep Mutalik unpacks the magic and math of how to win games when your opponent goes first…
Most games that pit two players or teams against each other require one of them to make the first play. This results in a built-in asymmetry, and the question arises: Should you go first or second?
Most people instinctively want to go first, and this intuition is usually borne out. In common two-player games, such as chess or tennis, it is a real, if modest, advantage to “win the toss” and go first. But sometimes it’s to your advantage to let your opponent make the first play.
In our February Insights puzzle, we presented four disparate situations in which, counterintuitively, the obligation to move is a serious and often decisive disadvantage. In chess, this is known as zugzwang — a German word meaning “move compulsion.”…
Four fascinating examples: “The Secrets of Zugzwang in Chess, Math and Pizzas,” from @PradeepMutalik.
* Fyodor Dostoyevsky, Notes from Underground
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As we play to win, we might recall that it was on this date in 2011 that scientists involved in the OPERA experiment (a collaboration between CERN and the Laboratori Nazionali del Gran Sasso) mistakenly observed neutrinos appearing to travel faster than light. OPERA scientists announced the results with the stated intent of promoting further inquiry and debate. Later the team reported two flaws in their equipment set-up that had caused errors far outside their original confidence interval: a fiber optic cable attached improperly, which caused the apparently faster-than-light measurements, and a clock oscillator ticking too fast; accounting for these two sources of error eliminated the faster-than-light results. But even before the sources of the error were discovered, the result was considered anomalous because speeds higher than that of light in a vacuum are generally thought to violate special relativity, a cornerstone of the modern understanding of physics for over a century.
“Do I rue a life wasted doing crosswords? Yes, but I do know the three-letter word for ‘regret'”*…
Efforts to diversify the crossword puzzle industry might be having the opposite effect. As Matt Hartmann explains, although puzzles are an increasingly important part of The New York Times’ and others’ business strategies, only a handful of people actually make a living from crosswords…
The conspiracy theory writes itself. Start looking, and you’ll notice how many New York Times crossword puzzles are co-constructed (the preferred term for what most people would refer to as co-written) by a professional crossword constructor and someone with a day job—it’s hard not to see all the artists, web developers, professors, and other titles that imply a degree of wealth and elite connections. As the pandemic handed the work-from-home class extra time for their hobbies, the number of first-timers published in the Times has skyrocketed. Obviously, rich people are paying others to get the glory of their name in ink.
But the theory is almost diametrically wrong. It turns out the crossword industry really does consist of earnest wordplay lovers donating their time to unpaid mentorships, generally as part of an industry-wide effort to bring new and underrepresented people into crosswords.
Unfortunately, the end result might be even more exclusive than a pay-to-play scheme. And a game that brings the Times at least one million monthly subscribers—at $1.25 a week or $40 for a year—provides a sustainable living wage for shockingly few people…
Learn why at “Inside the Elite, Underpaid, and Weird World of Crossword Writers,” from @themhartman in @newrepublic.
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As we fill in the blanks, we might recall that it was on this date in 1977 that South Park premiered on Comedy Central– where it runs to this day. The animated saga of Stan, Kyle, Eric, and Kenny and their exploits in their (titular) Colorado hometown has won five Emmys and a Peabody Award. A theatrical film, South Park: Bigger, Longer & Uncut, was released in June, 1999 to commercial and critical success, and scored an Academy Award nomination.
“Then we got into a labyrinth, and when we thought we were at the end, came out again at the beginning, having still to seek as much as ever.”*…
On the heels of Wordle‘s extraordinary success, there have been a rash of variations: e.g., Crosswordle, Absurdle, Quordle, even the NSFW Lewdle.
Now for the National Gallery of Art, another nifty puzzle: Artle.
Enjoy!
* Plato, Euthydemus
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As we play, we might recall that it was on this date in 1934 that Mandrake the Magician first appeared in newspapers. A comic strip, it was created by Lee Falk (before he created The Phantom)… and thus its crime-fighting, puzzle-solving hero is regarded by most historians of the form to have been America’s first comic superhero.
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