Latest news with #mathematicians
Daily Mail
4 days ago
- Science
- Daily Mail
Easy-looking math sum leaves people confused - can you solve it without a calculator?
Every so often it's good to exercise your brain with a math problem or two that forces you to recall principles you learned decades prior. As elementary as it might feel, you'd be surprised how easy it is to be stumped by a seemingly simple equation. This problem, posted on X, in particular may be challenging for even the best of mathematicians. Can you solve it without a calculator? 5+5 x 5+5 At first it may look easy, but it's important to go back to basics before diving in head first. The acronym PEMDAS can be handy when solving a sum like this one. Its letters spell out the order in which you should solve complex equations. The 'p' stands for parenthesis, then exponents, multiplication and division, and then adding or subtracting. Now, give it a try for yourself. Using the rules of PEMDAS, the first step is to solve the sum at the center of the problem, no matter how unnatural it may feel: 5 x 5 = 25 Now, the problem reads: 5+25+5 With simple addition from left to right, the final sum is easy: 5+25=30 30+5= 35 How did you do? If you couldn't solve it, don't worry, there are a few common mistakes that could have led to a different answer. The first, is adding on both sides before multiplying in the middle. Adding first on the left: 5+5=10 Making the problem: 10x5+5 Then, adding on the right: 5+5=10 That makes the solution: 10x10=100 Unfortunately, that method totally ignores the rules of PEMDAS, rendering the answer incorrect. Another easy error would be to solve from left to right, which may feel natural at first. 5+5=10 That leaves: 10x5+5 Then: 10x5=50 Which would make the solution: 50+5=55 But PEMDAS is a fool-proof way to guarantee the right answer every time and tease your brain with all-too-familiar math tricks.
New York Times
6 days ago
- General
- New York Times
Test Your Math Knowledge
'Math, Revealed,' our four-part series exploring the mathematics behind everyday objects and experiences, recently came to a smashing conclusion. One installment took a spin through 'taxicab geometry,' a wacky but vital corner of mathematics in which circles aren't round and pi equals 4. Another journey began with apples and pentagrams and led to Leonardo da Vinci, the golden ratio and ideal positioning of belly buttons. We had great fun on this adventure and hope to resume it before too long. In the meantime, here's a quiz to test what you learned and your general math knowledge. Enjoy! 1. In taxicab geometry, circles don't look round — they form sharp, angular shapes. What shape do they resemble? A triangle A diamond A hexagon A star An octagon 2. In taxicab geometry, even the value of pi is a surprise. What is it? About 3.14 About 1.41 Exactly 2 Exactly 3 Exactly 4 3. Mathematicians have long been fascinated by a special number that describes self-similar proportions. What is the approximate value of this 'golden ratio'? 1.41 1.50 1.62 1.75 2.00 4. Leonardo da Vinci's 'Vitruvian Man' has been analyzed endlessly for hidden patterns. According to a 2015 study, does the navel divide the figure according to the golden ratio? Yes, almost exactly. Yes, but only approximately. No, it follows a 2:1 ratio. No, it follows a 3:2 ratio. No, but the golden ratio is mentioned in Leonardo's notes. 5. In the densest possible arrangement of soda cans standing on a flat surface, each can in the middle touches the same number of neighbors. How many is it? Three Four Five Six Eight 6. In 2022, a mathematician won a Fields Medal for solving a problem about how to pack spheres tightly in eight dimensions. Who did it? Terence Tao Maryna Viazovska Thomas Hales Maryam Mirzakhani Ingrid Daubechies 7. The first four triangular numbers are 1, 3, 6 and 10. Why is 10 considered a triangular number? It's divisible by 3. It's the sum of three primes. It appears in the Pythagorean theorem. You can arrange 10 dots in an equilateral triangle. It's shaped like a triangle on the number line. 8. In 1672, Gottfried Wilhelm Leibniz found a clever way to add the reciprocals of all the triangular numbers. What sum did he get? 1 + 1/3 + 1/6 + 1/10 + ... 2 3 e Pi Infinity 9. At a wedding reception, the bride seats eight of her ex-boyfriends together at a table. (This actually happened to me once.) If each ex-boyfriend shakes hands with each of the others, how many handshakes occur in total? 28 32 36 40 56 10. Four bugs start in the corners of a square that measures 1 foot by 1 foot. Each bug chases its clockwise neighbor, always crawling directly toward the neighbor's current position. If they all crawl at the same speed, how far does each bug travel before they all meet at the center? Not enough information is given. I never took calculus. Help! 1 foot 2 feet 1.41 feet Questions and answers 1 through 8: photo illustrations by Jens Mortensen for The New York Times; answer 6: photo by Brendan Hoffman for The New York Times; question 9: photo illustration by The New York Times, source photo via Alamy; question 10: photo illustration by The New York Times, source photos by Balarama Heller for The New York Times. Produced by Alan Burdick, Alice Fang, Marcelle Hopkins and Matt McCann.
Daily Mail
08-07-2025
- Entertainment
- Daily Mail
Foolproof way to win any jackpot according to maths (but there is a teeny catch)
I have a completely foolproof, 100-per-cent-guaranteed method for winning any lottery you like. If you follow my very simple method, you will absolutely win the maximum jackpot possible. There is just one teeny, tiny catch – you're going to need to already be a multimillionaire, or at least have a lot of rich friends. Let's take the US Powerball lottery as an example. To play, you pick five different 'white' numbers from 1 to 69, and a sixth 'red' number from 1 to 26 – this last number can be a repeat of one of the white ones. How many different possible lottery tickets are there? To calculate that, we need to turn to a field of mathematics called combinatorics, which, as the name suggests, is a way of calculating the number of possible combinations of objects. Picking numbers from an unordered set, as with a lottery, is an example of an 'n choose k' problem, where n is the total number of objects we can choose from (69 in the case of the white Powerball numbers) and k is the number of objects we want to pick from that set. Crucially, because you can't repeat the white numbers, these choices are made 'without replacement' – as each winning numbered ball is selected for the lottery, it doesn't go back into the pool of available choices. Mathematicians have a handy formula for calculating the number of possible results of an n choose k problem: n! / (k! × (n – k)!). If you've not encountered it before, a mathematical '!' doesn't mean we're very excited – it's a symbol that stands for the factorial of a number, which is simply the number you get when you multiply a whole number, or integer, by all of those smaller than itself. For example, 3! = 3 × 2 × 1 = 6. Plugging in 69 for n and 5 for k, we get a total of 11,238,513. That's quite a lot of possible lottery tickets, but as we will see later on, perhaps not enough. This is where the red Powerball comes in – it essentially means you are playing two lotteries at once and must win both for the largest prize. This makes it a lot harder to win. If you just simply added a sixth white ball, you'd have a total of 119,877,472 possibilities. But because there are 26 possibilities for red balls, we multiply the combinations of the white balls by 26 to get a total of 292,201,338 – much higher. Ok, so we have just over 292 million possible Powerball tickets. Now, here comes the trick to always winning – you simply buy every possible ticket. Simple maybe isn't quite the right word here, given the logistics involved, and most importantly, with tickets costing $2 apiece, you will need to have over half a billion dollars on hand. Is it enough to absolutely guarantee a big payout? That's a slightly tricky question to answer. The Powerball jackpot rolls over each week it remains unclaimed, meaning the amount you can win varies. But there have been only 15 examples of the jackpot getting higher than the $584 million you need to buy every ticket, so most of the time it's not worth it. The profit in doing so is further diminished by the fact that jackpots can be shared by multiple winners who choose the same numbers, and that around 30 per cent of the winnings are taken in tax. In a way, none of this should be surprising – if there was a guaranteed way to win lotteries and make a profit, people would be doing it all the time and lottery runners would go bust. But surprisingly, badly designed lotteries do crop up – and savvy investors sweep in to make a killing. One of the first examples of this kind of lottery busting involved the writer and philosopher Voltaire. Together with Charles Marie de La Condamine, a mathematician, he formed a syndicate to buy all the tickets in a lottery linked to French government debt. Exactly how he went about this is murky and there is some suggestion of skullduggery, such as not having to pay full price for the tickets, but the upshot is that the syndicate appears to have won repeatedly before the authorities shut the lottery down in 1730. Writing about it later, in the third person, Voltaire said 'winning lots were paid in cash and all in such a way that any group of people who had bought all the tickets stood to win a million francs. Voltaire entered into association with numerous company and struck lucky.' More modern lotteries have also suffered the same fate. A famous example is the Irish National Lottery, which was bought out by a syndicate of a couple dozen people in 1992. At the time, players had to pick six numbers from 1 to 36, which our n choose k formula tells us produces 1,947,792 possible tickets. With tickets costing 50 Irish pence (the currency at the time), the conspirators raised the necessary £973,896 and began buying tickets for an estimated £1.7 million prize draw. Lottery organisers got wind of the scheme and began limiting the number of tickets each vendor could sell, meaning the syndicate only managed to purchase about 80 per cent of ticket combinations. In the end, it shared the jackpot with two other winners, giving it a loss-making prize of £568,682. Luckily for the syndicate, the lottery had also introduced a guaranteed £100 prize for matching four numbers, which brought its total to £1,166,000. The Irish National Lottery quickly changed the rules to avoid a similar scheme, and these days requires six numbers chosen from 47, upping the number of tickets to 10,737,573. The jackpot is capped at €18.9 million, while tickets are €2 each, ensuring that buying the lottery will never be profitable. Despite the fact that the risks of a poorly designed lottery should now be well understood, these incidents may still be occurring. One extraordinary potential example came in 2023, when a syndicate won a $95 million jackpot in the Texas State Lottery. The Texas lottery is 54 choose 6, a total of 25,827,165 combinations, and tickets cost $1 each, making this a worthwhile enterprise – but the syndicate may have had assistance from the lottery organisers themselves. While the fallout from the scandal is still unfolding, and it is not known whether anything illegal has occurred, the European-based syndicate, working through local retailers, may have acquired ticket-printing terminals from the organisers of the Texas lottery, allowing it to purchase the necessary tickets and smooth over the logistics. The lottery commissioner at the time has denied being part of any illegal scheme. And no criminal charges have been filed – the lawyer that represents the syndicate that claimed the jackpot, known as Rook TX, said 'All applicable laws, rules and regulations were followed.' So there you have it. Provided that you have a large sum of upfront cash, and can find a lottery where the organisers have failed to do their due diligence with the n choose k formula, you can make a tidy profit. Good luck!
Daily Mail
07-07-2025
- General
- Daily Mail
Grade school sum looks easy but leaves people confused - can you solve it?
A grade school arithmetic problem has left people scratching their heads. The problem posted by @BholanathDutta on X read: 4+4x4+4 Before you jump into the deep end too quickly and try to find the solution, first remind yourself of simple math principles so you don't make a basic mistake. This sum is a simple lesson in mathematicians' favorite acronym: PEMDAS. PEMDAS is frequently taught as a way for mathletes to remember the correct order in which to solve compound math equations. It stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. There are no parentheses or exponents in this problem so multiplication comes first. It may feel unnatural to solve from the center, but give it a go and see if you can find the sum in 30 seconds or less. 4 x 4 = 16 With one step down, the problem reads: 4+16+4 From there, it's just simple addition: 4+16= 20 Now we have: 20+4 = 24 Were you able to solve it? There are a couple of easy errors that other wannabe mathematicians fell victim to while trying to get to the bottom of things. The first is solving the equation left to right, instead of implementing PEMDAS. 4+4=8 8 x 4 = 32 32 + 4 = 36 Another easy mistake to make is putting the addition first. By adding 4+4 on both sides of the equation first the simplified problem would become: 8 x 8 = 64 Unfortunately, this mixes up the reliable acronym PEMDAS and leaves the solution unsolved. How did you do? Were you able to rely on math principles learned decades ago or did you too fall victim to one of these easy errors?
Yahoo
20-06-2025
- Science
- Yahoo
Mathematicians discover a completely new way to find prime numbers
When you buy through links on our articles, Future and its syndication partners may earn a commission. For centuries, prime numbers have captured the imaginations of mathematicians, who continue to search for new patterns that help identify them and the way they're distributed among other numbers. Primes are whole numbers that are greater than 1 and are divisible by only 1 and themselves. The three smallest prime numbers are 2, 3 and 5. It's easy to find out if small numbers are prime — one simply needs to check what numbers can factor them. When mathematicians consider large numbers, however, the task of discerning which ones are prime quickly mushrooms in difficulty. Although it might be practical to check if, say, the numbers 10 or 1,000 have more than two factors, that strategy is unfavorable or even untenable for checking if gigantic numbers are prime or composite. For instance, the largest known prime number, which is 2136279841 − 1, is 41,024,320 digits long. At first, that number may seem mind-bogglingly large. Given that there are infinitely many positive integers of all different sizes, however, this number is minuscule compared with even larger primes. Furthermore, mathematicians want to do more than just tediously attempt to factor numbers one by one to determine if any given integer is prime. "We're interested in the prime numbers because there are infinitely many of them, but it's very difficult to identify any patterns in them," says Ken Ono, a mathematician at the University of Virginia. Still, one main goal is to determine how prime numbers are distributed within larger sets of numbers. Recently, Ono and two of his colleagues — William Craig, a mathematician at the U.S. Naval Academy, and Jan-Willem van Ittersum, a mathematician at the University of Cologne in Germany — identified a whole new approach for finding prime numbers. "We have described infinitely many new kinds of criteria for exactly determining the set of prime numbers, all of which are very different from 'If you can't factor it, it must be prime,'" Ono says. He and his colleagues' paper, published in the Proceedings of the National Academy of Sciences USA, was runner-up for a physical science prize that recognizes scientific excellence and originality. In some sense, the finding offers an infinite number of new definitions for what it means for numbers to be prime, Ono notes. At the heart of the team's strategy is a notion called integer partitions. "The theory of partitions is very old," Ono says. It dates back to the 18th-century Swiss mathematician Leonhard Euler, and it has continued to be expanded and refined by mathematicians over time. "Partitions, at first glance, seem to be the stuff of child's play," Ono says. "How many ways can you add up numbers to get other numbers?" For instance, the number 5 has seven partitions: 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1 and 1 + 1 + 1 + 1 + 1. Yet the concept turns out to be powerful as a hidden key that unlocks new ways of detecting primes. "It is remarkable that such a classical combinatorial object — the partition function — can be used to detect primes in this novel way," says Kathrin Bringmann, a mathematician at the University of Cologne. (Bringmann has worked with Ono and Craig before, and she's currently van Ittersum's postdoctoral adviser, but she wasn't involved with this research.) Ono notes that the idea for this approach originated in a question posed by one of his former students, Robert Schneider, who's now a mathematician at Michigan Technological University. Ono, Craig and van Ittersum proved that prime numbers are the solutions of an infinite number of a particular type of polynomial equation in partition functions. Named Diophantine equations after third-century mathematician Diophantus of Alexandria (and studied long before him), these expressions can have integer solutions or rational ones (meaning they can be written as a fraction). In other words, the finding shows that "integer partitions detect the primes in infinitely many natural ways," the researchers wrote in their PNAS paper. George Andrews, a mathematician at Pennsylvania State University, who edited the PNAS paper but wasn't involved with the research, describes the finding as "something that's brand new" and "not something that was anticipated," making it difficult to predict "where it will lead." Related: What is the largest known prime number? The discovery goes beyond probing the distribution of prime numbers. "We're actually nailing all the prime numbers on the nose," Ono says. In this method, you can plug an integer that is 2 or larger into particular equations, and if they are true, then the integer is prime. One such equation is (3n3 − 13n2 + 18n − 8)M1(n) + (12n2 − 120n + 212)M2(n) − 960M3(n) = 0, where M1(n), M2(n) and M3(n) are well-studied partition functions. "More generally," for a particular type of partition function, "we prove that there are infinitely many such prime detecting equations with constant coefficients," the researchers wrote in their PNAS paper. Put more simply, "it's almost like our work gives you infinitely many new definitions for prime," Ono says. "That's kind of mind-blowing." The team's findings could lead to many new discoveries, Bringmann notes. "Beyond its intrinsic mathematical interest, this work may inspire further investigations into the surprising algebraic or analytic properties hidden in combinatorial functions," she says. In combinatorics — the mathematics of counting — combinatorial functions are used to describe the number of ways that items in sets can be chosen or arranged. "More broadly, it shows the richness of connections in mathematics," she adds. "These kinds of results often stimulate fresh thinking across subfields." Bringmann suggests some potential ways that mathematicians could build on the research. For instance, they could explore what other types of mathematical structures could be found using partition functions or look for ways that the main result could be expanded to study different types of numbers. "Are there generalizations of the main result to other sequences, such as composite numbers or values of arithmetic functions?" she asks. "Ken Ono is, in my opinion, one of the most exciting mathematicians around today," Andrews says. "This isn't the first time that he has seen into a classic problem and brought really new things to light." RELATED STORIES —Largest known prime number, spanning 41 million digits, discovered by amateur mathematician using free software —'Dramatic revision of a basic chapter in algebra': Mathematicians devise new way to solve devilishly difficult equations —Mathematicians just solved a 125-year-old problem, uniting 3 theories in physics There remains a glut of open questions about prime numbers, many of which are long-standing. Two examples are the twin prime conjecture and Goldbach's conjecture. The twin prime conjecture states that there are infinitely many twin primes — prime numbers that are separated by a value of two. The numbers 5 and 7 are twin primes, as are 11 and 13. Goldbach's conjecture states that "every even number bigger than 2 is a sum of two primes in at least one way," Ono says. But no one has proven this conjecture to be true. "Problems like that have befuddled mathematicians and number theorists for generations, almost throughout the entire history of number theory," Ono says. Although his team's recent finding doesn't solve those problems, he says, it's a profound example of how mathematicians are pushing boundaries to better understand the mysterious nature of prime numbers. This article was first published at Scientific American. © All rights reserved. Follow on TikTok and Instagram, X and Facebook.
