Latest news with #MartinGardner
Hindustan Times
21-07-2025
- Entertainment
- Hindustan Times
One for you, one for me
We learn some properties of numbers early in life, such as all products of 9 having a digital root of 9, and all single-digit multiplications with 37 leading to products with three repeating digits. Other properties may pass unnoticed for years until they are presented to us in the face, and we wonder why we hadn't noticed it all along. Representational image.(Shutterstock) The following puzzle exploits a property I had not noticed until I first read and solved the puzzle some three decades ago. Martin Gardner was particularly fond of it, and credits it to two readers who brought it to his notice separately. My version tweaks the original described by Garden. The puzzle may seem to lack enough information when you first read it, but think a bit and you will find it is as easy as it is enjoyable. #Puzzle 152.1 A publisher hands a number of copies of a school textbook to two vendors. The deal is that the book is to be sold at its cover price, with the vendors receiving a commission based on the number of copies they sell. By coincidence, the number of copies sold turns out to match the commission in rupees per book. That is to say, if the commission is x rupees per book, then they have sold exactly x copies. The publisher pays them the commission in ₹10 notes and ₹1 coins, the number of coins being less than 10. Having no other cash in their pockets, the vendors decide to share the notes and coins in the style of Blondie and Tuco in The Good, The Bad and The Ugly. 'One for you, one for me,' Vendor #1 says, beginning to distribute the ₹10 notes. 'One for you, one for me…' And so on, until the last note is reached: 'One for you.' It strikes Vendor #1 that he has handed Vendor#2 the first note as well as the last. 'Hey, you got ₹10 more than I did because the number of notes was odd.' 'Never mind,' says Vendor #2, 'you keep all the ₹1 coins.' 'But that's less than 10 coins and so less than ₹10. You still end up with a higher share of our commission,' says Vendor #1. 'Never mind,' Vendor #2 repeats himself. 'Let me pay you the difference.' He opens his UPI app, and makes the transaction. 'There, we now have equal shares.' How much does Vendor #2 send Vendor #1 by UPI? #Puzzle 152.2 A pet shop manager assures a customer that the parrot he is offering will repeat every word she hears. The customer tries to check this out but the parrot is fast asleep, having been drugged by the manager. To make sure he is not cheated, the customer makes the manager put the assurance in writing: 'Guaranteed that Parrot #152.2 sold to [customer's name] in July 2025 will repeat every word she hears, failing which the payment of [amount] shall be returned to the customer.' Thus assured, the customer buys the parrot. At home, when the pet wakes up, the buyer says 'Hi!' No reply. He tries various other words, but not a word from the parrot. The buyer rushes angrily to the pet shop, but the manager refuses a refund claiming no terms have been breached. The buyer goes to the local don for arbitration. To his disappointment, the don rules in the manager's favour. Explain why the manager hadn't bluffed. MAILBOX: LAST WEEK'S SOLVERS #Puzzle 151.1 Hi Kabir, Here is how the mathematical trick with cards works. The digital roots of the numbers from 43 to 51 are: (43 = 7), (44 = 8), (45 = 9), (46 = 1), (47 = 2), (48 = 3), (49 = 4), (50 = 5), (51 = 6). We can see that the sequence of digital roots matches the sequence of cards 7-8-9-1-2-3-4-5-6. Whatever the number of cards added to the pile of 43 cards, the new digital root will match the card that is not transferred and turned face up. Also, note that the digital root of any number will remain the same even after splitting that number into two or more parts and adding the digital roots of those parts. — Shishir Gupta, Indore #Puzzle 151.2 Hi Kabir, The four cards are, from left to right, are: Queen of Spades, Ace of Hearts, Jack of Clubs, King of Diamonds. — Aditya Krishnan, NMIMS Mumbai Solved both puzzles: Shishir Gupta (Indore), Aditya Krishnan (NMIMS Mumbai), Vinod Mahajan (Delhi), Dr Sunita Gupta (Delhi), Anil Khanna (Ghaziabad), YK Munjal (Delhi),Sanjay Gupta (Delhi), Professor Anshul Kumar (Copenhagen), Ajay Ashok (Delhi), Yadvendra Somra (Sonipat) Solved Puzzle 151.2: Dr Vivek Jain (Baroda) Problematics will be back next week. Please send in your replies by Friday noon to problematics@
Hindustan Times
14-07-2025
- Entertainment
- Hindustan Times
The roots of a card trick
By a happy coincidence, just weeks after we discussed digital roots and how they can help in determining if a number is a perfect square, I came across the concept again. To recap, the digital root is the result of repeatedly adding the digits of a number until you are left with a single digit. For example, the digital root of 2697 is obtained by adding 2 + 6 + 9 + 7 = 24, then 2 + 4 = 6. Welcome to Problematics! (Shutterstock) The late American puzzler and author Martin Gardner describes a card trick based on digital roots. The trick was invented by Stewart James, a magician from Ontario. It is very easy to see why it works, but deducing that should be fun. You may also enjoy playing the trick on an audience. #Puzzle 151.1 From a standard deck of 52, a magician selects and separates nine cards, and orders them in the following sequence: ace (top)-2-3-4-5-6-7-8-9 (bottom). He shows the audience these cards, telling them the ace represents 1. Now holding them face down, the magician pretends to cut the nine cards randomly, but what he actually does is bring the bottom three cards to the top. The order, which is not shown to the audience, is now 7 (top)-8-9-1-2-3-4-5-6 (bottom). The magician now places the pile of nine cards and the remainder of the deck side by side, and invites a member of the audience on stage. 'I will transfer cards, one by one, from the pile of nine to the main deck. You may select any card being transferred, calling out 'Stop!', at which stage I shall stop the transfer and display the selected card,' he tells the spectator. Just to make sure there is no misunderstanding, let's take an illustrative example. The magician starts to transfer cards from the pile of nine to the main deck of 43. Let us say he has transferred four cards (7-8-9-1) when the spectator says 'Stop!' on the fifth card. The magician turns up the fifth card, which is 2. He places the 2 face-up on the remainder of the pile (the cards that have not been transferred). The magician invites two more spectators. One of them is asked to cut the deck (the set not containing the face-up card) into two new piles, randomly. Let us say the spectator cuts the deck (which now has 47 cards after the transfer) into piles of 18 and 29 cards. The third spectator is asked to count the cards in each pile, to add these digits until she is left with a single digit. She adds as follows: 1 + 8 = 9, and 2 + 9 = 11, then 1 + 1 = 2. She then adds these two roots to get 9 + 2 = 11, then 1 + 1 = 2.. Bingo! The result matches the face-up card selected by the first spectator. To reiterate, the above is only an illustrative example. No matter which card the first spectator selects, and no matter how the second spectator splits the deck into two piles, the prescribed mathematical steps will always lead to a digital root that is the same as the selected card. What is the mathematical principle that makes the trick work? #Puzzle 151.2 An ace, a king, a queen and a jack are taken from a standard deck of cards. Each of them belongs to a different suit, so that all four suits are represented. They are now laid side by side in a row. Your clues: The king is to the right of the club. The club is to the right of the ace. The spade is to the left of the ace. The heart is to the left of the jack. The diamond is to the right of the jack. Name the four cards in order from left to right. MAILBOX: LAST WEEK'S SOLVERS #Puzzle 150.1 Dear Kabir, The first puzzle last week seemed tough at first, but I was able to solve it easily after I wrote down the prime factors of 19800, grouped these factors in all possible combinations and rearranged the corresponding letters. The two possible names of Bloke's wife are: DIVYA and VIDYA. — Y K Munjal, Delhi #Puzzle 150.2 Hi Kabir, The two given statements are paradoxical — each statement contradicts itself. — Professor Anshul Kumar, Copenhagen *** Another example of an invalid statement is: "I am a liar", which if true, would nullify me being a liar, and if false, would indicate that I am lying. — Sampath Kumar V, Coimbatore Apologies to Yadvendra Somra for omitting his name last week. He had correctly solved Puzzle 149.1 Solved both puzzles: Y K Munjal (Delhi), Professor Anshul Kumar (Copenhagen), Sampath Kumar V (Coimbatore), Yadvendra Somra (Sonipat), Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Dr Sunita Gupta (Delhi), Shri Ram Aggarwal (Delhi), Abhinav Mital (Singapore), Ajay Ashok (Delhi), Sabornee Jana (Mumbai), Shishir Gupta (Indore), Aditya Krishnan (NMIMS Mumbai) Solved #Puzzle 150.1: Anil Khanna (Ghaziabad) Solved #Puzzle 150.2: Dr Vivek Jain (Baroda) Problematics will be back next week. Please send in your replies by Friday noon to problematics@
Scientific American
16-06-2025
- Science
- Scientific American
Truly Intelligent AI Could Play by the Rules, No Matter How Strange
Tic-tac-toe is about as simple as games get—but as Scientific American 's legendary contributor Martin Gardner pointed out almost 70 years ago, it has complex variations and strategic aspects. They range from 'reverse' games—where the first player to make three in a row loses—to three-dimensional versions played on cubes and beyond. Gardner's games, even if they boggle a typical human mind, might point us to a way to make artificial intelligence more humanlike. That's because games in their endless variety—with rules that must be imagined, understood and followed—are part of what makes us human. Navigating rules is also a key challenge for AI models as they start to approximate human thought. And as things stand, it's a challenge where most of these models fall short. That's a big deal because if there's a path to artificial general intelligence, the ultimate goal of machine-learning and AI research, it can only come through building AIs that are capable of interpreting, adapting to and rigidly following the rules we set for them. On supporting science journalism If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today. To drive the development of such AI, we must develop a new test—let's call it the Gardner test—in which an AI is surprised with the rules of a game and is then expected to play by those rules without human intervention. One simple way to achieve the surprise is to disclose the rules only when the game begins. The Gardner test, with apologies to the Turing test, is inspired by and builds on the pioneering work in AI on general game playing (GGP), a field largely shaped by Stanford University professor Michael Genesereth. In GGP competitions, AIs running on standard laptops face off against other AIs in games whose rules—written in a formal mathematical language —are revealed only at the start. The test proposed here advances a new frontier: accepting game rules expressed in a natural language such as English. Once a distant goal, this is now within reach of modern AIs because of the recent breakthroughs in large language models (LLMs) such as those that power ChatGPT and that fall within the families of Claude and Llama. The proposed challenge should include a battery of tests that could be initially focused on games that have been staples of GGP competitions such as Connect Four, Hex and Pentago. It should also leverage an impressive array of games that Gardner wrote about. Test design could benefit from the involvement of the vibrant international GGP research community, developers of frontier AI models and, of course, diehard Martin Gardner fans. But to pass the new test, it isn't enough to create an AI system that's good at playing one specific predetermined game or even many. Instead, an AI must be designed to master any strategy game on the fly. Strategy games require humanlike ability to think across and beyond multiple steps, deal with unpredictable responses, adapt to changing objectives and still conform to a strict rule set. That's a big leap from today's top game-playing AI models, which rely on knowing the rules in advance to train their algorithms. Consider, for instance, AlphaZero, the revolutionary AI model that's capable of playing three games—chess, Go and shogi (Japanese chess)—at a superhuman level. AlphaZero learns through a technique known as 'self-play'—it repeatedly plays against a copy of itself, and from that experience, it gets better over time. Self-play, however, requires the rules of each game to be set before training. AlphaZero's ability to master complex games is undoubtedly impressive, but it's a brittle system: if you present AlphaZero with a game different than the ones it's learned, it will be completely flummoxed. In contrast, an AI model performing well on the proposed new test would be capable of adapting to new rules, even in the absence of data; it would play any game and follow any novel rule set with power and precision. That last point—precision—is an important one. You can prompt many generative AI systems to execute variants on simple games, and they'll play along: ChatGPT can play a 4×4 or 5×5 variant of tic-tac-toe, for instance. But an LLM prompt is best thought of as a suggestion rather than a concrete set of rules—that's why we often have to coax, wheedle and prompt tune LLMs into doing exactly what we want. A general intelligence that would pass the Gardner test, by contrast, would by definition be able to follow the rules perfectly: not following a rule exactly would mean failing the test. Specialized tools that operate without truly understanding the rules tend to color outside the lines, reproducing past errors from training data rather than adhering to the rules we set. It's easy to imagine real-world scenarios in which such errors could be catastrophic: in a national security context, for instance, AI capabilities are needed that can accurately apply rules of engagement dynamically or negotiate subtle but crucial differences in legal and command authorities. In finance, programmable money is emerging as a new form of currency that can obey rules of ownership and transferability—and misapplying these rules could lead to financial disaster. Ironically, building AI systems that can follow rules rigorously would ultimately make it possible to create machine intelligences that are far more humanlike in their flexibility and ability to adapt to uncertain and novel situations. When we think of human game players, we tend to think of specialists: Magnus Carlsen is a great chess player but might not be so hot at Texas Hold'Em. The point, though, is that humans are capable of generalizing; if Carlsen ever gave up chess, he could be a decent contender for the Pentamind World Championship, which celebrates the best all-round games player. Game playing with a novel set of rules is crucial to the next evolution of AI because it will potentially let us create AIs that will be capable of anything—but that will also meticulously and reliably follow the rules we set for them. If we want powerful but safe AI, testing its ability in playing games on the fly might be the best path forward.
