Stories with Tag challenge

• Making any integer with four 2s

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  Posted: 2025-02-23 02:25:51

  Verbose tl;dr

  The post explores the mathematical puzzle of representing any integer using four twos and a limited set of operations. It demonstrates how combining operations like addition, subtraction, multiplication, division, square roots, factorials, decimal points, and concatenation, along with concepts like double factorials and the gamma function (a generalization of the factorial), allows for creative expression of numerous integers. While acknowledging the potential for more complex representations using less common operations, the post focuses on showcasing the flexibility and surprising reach of this mathematical exercise using a relatively small toolkit of functions. It ultimately highlights the challenge and ingenuity involved in manipulating a limited set of numbers to achieve a wide range of results.

  This blog post delves into the fascinating mathematical puzzle of representing any given integer using exactly four instances of the digit '2', along with a variety of permitted mathematical operations. The author meticulously outlines the permissible operations, which include the fundamental arithmetic operations of addition, subtraction, multiplication, and division. Furthermore, the author allows for the use of exponentiation, square roots (represented by the conventional radical symbol), factorials, concatenation (joining the digits together, e.g., forming 22 from two 2s), and decimal points (to create numbers like 2.2). Parentheses are also permitted to explicitly control the order of operations and enforce desired groupings.

  The central challenge, as articulated by the author, is to devise expressions using precisely four 2s and the allowed operations to generate each integer sequentially. The post then proceeds to demonstrate solutions for a range of integers, starting from 0 and continuing upwards. These demonstrations showcase the ingenious application of various combinations of the allowed operations. For example, the integer 0 is represented as (2 + 2 - 2 - 2). The author highlights the escalating complexity involved in representing larger integers, illustrating how cleverly combining operations becomes crucial to achieve the desired results.

  The post emphasizes the explorative and often counterintuitive nature of the problem-solving process. It highlights the satisfaction derived from discovering elegant and sometimes unexpected solutions. The author does not claim to provide a generalized formula or algorithm for solving the puzzle for any given integer. Rather, the post focuses on showcasing specific examples and demonstrating the kind of creative mathematical thinking required to tackle such challenges. The ultimate objective isn't necessarily to find all possible representations for each integer, but rather to demonstrate the feasibility of representing a variety of integers within the imposed constraints of using only four 2s and the specified operations.

  • mathematics
  • puzzles
  • number theory
  • Arithmetic
  • integers
  • recreational mathematics
  • Problem Solving
  • four twos
  • 2s
  • challenge

  Summary of Comments ( 140 )
  https://news.ycombinator.com/item?id=43145753

  Verbose tl;dr

  Hacker News users generally enjoyed the puzzle presented in the linked article about constructing integers using four twos. Several commenters explored alternative solutions using different mathematical operations like bitwise XOR, square roots, and logarithms, showcasing a playful engagement with the challenge. Some discussed the arbitrary nature of the "four twos" constraint, suggesting that similar puzzles could be devised with other numbers or constraints. A few comments delved into the role of such puzzles in education, highlighting their value in encouraging creative problem-solving. One commenter pointed out the similarity to the "four fours" puzzle, referencing a website dedicated to exploring its variations.

  The Hacker News post "Making any integer with four 2s" has generated several comments discussing various aspects of the mathematical puzzle presented in the linked article. The comments largely revolve around alternative solutions, generalizations of the puzzle, and debates about the "legality" of certain operations.

  One commenter points out that the puzzle allows for a broader range of operations than typically intended, making it relatively easy to generate any integer. They specifically mention the use of the square root and floor function, which allows for the creation of 1 (sqrt(2)/sqrt(2) = 1) and 0 (floor(sqrt(2)/22) = 0), and from there any other integer. This commenter also notes the lack of strict rules surrounding concatenation, which can be exploited further.

  Another commenter focuses on the use of the logarithm function, suggesting a more generalized approach. They propose using log base 2 of the square root of 2 (log₂(√2)) to obtain 1/2, and subsequently generating other fractions and integers. This comment sparks a discussion about which operations are "acceptable" within the puzzle's framework, with some arguing that logarithms are too powerful and stretch the intended spirit of the challenge.

  The discussion also touches upon the concept of using the successor function, denoted as S(n) = n+1, though this is generally dismissed as being outside the usual scope of such puzzles. A similar sentiment is expressed regarding the use of infinite series, which are considered too powerful and make the puzzle trivial.

  Several commenters express their enjoyment of the puzzle and explore alternative solutions using more standard operations like addition, subtraction, multiplication, division, square roots, and factorials. Some commenters propose restricting the allowed operations to make the puzzle more challenging.

  A significant part of the discussion centers on the ambiguity of the rules, specifically regarding concatenation and the allowed set of operations. This ambiguity leads to creative solutions but also to debates on whether those solutions adhere to the (unstated) intended rules of the puzzle. Some commenters suggest that explicitly defining the allowed operations would lead to a more focused and interesting challenge.

• The Secret Poker Game You Can Play on the Subway

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  Posted: 2025-02-18 10:37:55

  Verbose tl;dr

  "Subway Poker" is a silent, observational game played by mentally assigning poker hands to fellow subway passengers based on the first five visible digits of their MetroCards. Players cannot influence the cards they are "dealt" and simply observe the hands around them, comparing their imagined hands to others' to determine a winner. The game offers a discreet and engaging way to pass the time during a commute, adding a layer of amusement to the mundane act of people-watching.

  Within the bustling, subterranean world of the metropolitan subway system, where weary commuters are often lost in their own thoughts or engrossed in digital distractions, lies a hidden realm of clandestine amusement: a silent, subtle game of poker, played not with cards and chips, but with the diverse tapestry of human faces that populate these underground carriages. This ingeniously conceived pastime, detailed in the aforementioned blog post, transforms the mundane commute into an engaging exercise in observation and deduction, offering a welcome respite from the monotony of the daily transit grind.

  The rules of this unspoken game are elegantly simple, yet offer a surprising depth of strategic potential. Each participant, without overtly acknowledging their involvement, selects a hand of five "cards" represented by the first five individuals they observe exhibiting a specific, pre-determined characteristic. This characteristic could be anything visually discernible, from the color of a passenger's hat to the presence of a particular accessory, thereby adding an element of personalization to the game. The chosen characteristic serves as the suit, and the specific manifestation of that characteristic within each observed individual acts as the card's rank. For instance, if the chosen characteristic is "type of footwear," a player might observe individuals wearing sneakers, boots, sandals, loafers, and slippers, forming their five-card hand.

  Following the established conventions of traditional poker, the objective is to assemble the highest-ranking hand possible based on the standard hierarchy of poker hands, from a high card to a royal flush. The inherent challenge, and indeed the source of much of the game's amusement, lies in the unpredictable nature of the "cards" dealt by the ever-shifting population of the subway car. The constant flux of passengers boarding and disembarking introduces an element of dynamism, as the player's hand can evolve with each new arrival, offering both opportunities for improvement and the potential for agonizing near-misses. The game concludes when the player reaches their destination, at which point they can mentally evaluate their final hand and, should they choose, discreetly revel in their subterranean poker prowess. The beauty of this silent game lies in its inherent privacy; it is a solitary pursuit, played entirely within the confines of one's own mind, requiring no outward interaction with fellow passengers, thereby preserving the unspoken code of conduct of the subway environment. This subtle game transforms the daily commute from a passive experience into an active engagement with the surrounding human landscape, offering a unique and entertaining way to pass the time beneath the city streets.

  • poker
  • subway
  • Game
  • travel
  • Commute
  • Entertainment
  • people watching
  • observation
  • social interaction
  • nonverbal communication
  • Psychology
  • secret
  • hidden
  • subtle
  • City Life
  • urban
  • discrete
  • challenge

  Summary of Comments ( 0 )
  https://news.ycombinator.com/item?id=43088124

  Verbose tl;dr

  Commenters on Hacker News largely enjoyed the subway poker concept, calling it "charming" and "delightful." Some discussed strategies, like focusing on suits rather than specific card values for easier memorization, or intentionally losing rounds to mislead other players. A few debated the feasibility of the game in real-world scenarios, citing crowded subways and the difficulty of maintaining eye contact. Several expressed interest in trying it themselves, while others suggested variations, like using phone apps or playing with larger groups. Some reminisced about similar silent games played in childhood. The overall sentiment was positive, with the game seen as a fun and engaging way to pass the time on public transit.

  The Hacker News post titled "The Secret Poker Game You Can Play on the Subway" spawned a moderate discussion with a few interesting threads. Several commenters focused on the practicality and limitations of the proposed game.

  One commenter pointed out the difficulty of maintaining eye contact, a core mechanic of the game, in a crowded subway environment, especially considering cultural norms around sustained eye contact with strangers. They also suggested the game might be more enjoyable and less awkward in a setting with a smaller, more static group of people, like a bus.

  Another commenter highlighted the potential for misinterpretation and unintended social consequences, especially in a diverse urban environment. They noted that attempting to initiate the game could be mistaken for aggressive behavior or unwanted attention, leading to uncomfortable or even dangerous situations. This concern was echoed by others who suggested that the game's reliance on subtle cues could easily be misunderstood.

  Several commenters questioned the actual "game" aspect, noting the lack of any real stakes or meaningful strategy beyond simple bluffing. One suggested it was more of a social experiment or a shared moment of silent acknowledgement than a true game. This sentiment led to a discussion about the nature of games and whether a competitive element is essential. Someone posited that the enjoyment derived from Subway Poker might stem from the shared secret and the subtle thrill of engaging in a clandestine activity.

  A technical point raised by one commenter was the difficulty of establishing a clear "winner" given the fleeting nature of subway interactions. They proposed a scoring system based on the duration of eye contact to add a more concrete competitive element.

  Finally, some commenters simply expressed amusement at the concept and shared their own experiences with similar unspoken interactions on public transport. One mentioned the "nod game," a simple exchange of nods between strangers, as another example of this type of subtle social interaction.

  Overall, the discussion focused on the practicality, social implications, and the very definition of the proposed "Subway Poker" game. While some embraced the idea with enthusiasm, others expressed skepticism and concern about its potential for misinterpretation and awkwardness.