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, decimals, and concatenation, alongside techniques like logarithms and the gamma function (a generalization of the factorial), allows for expressing a wide range of integers. The author showcases examples and discusses the challenges of representing larger numbers, particularly prime numbers, due to the increasing complexity of the required expressions. The ultimate goal isn't a formal proof, but rather a practical exploration of the expressive power of combining these mathematical tools with a limited set of starting digits.
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.
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.
Certain prime numbers possess aesthetically pleasing or curious properties that make them stand out and become targets for "prime hunters." These include palindromic primes (reading the same forwards and backwards), repunit primes (consisting only of the digit 1), and Mersenne primes (one less than a power of two). The rarity and mathematical beauty of these special primes drive both amateur and professional mathematicians to seek them out using sophisticated algorithms and distributed computing projects, pushing the boundaries of computational power and our understanding of prime number distribution.
HN commenters largely discussed the memorability and aesthetics of the listed prime numbers, debating whether the criteria truly made them special or just reflected pattern-seeking tendencies. Some questioned the article's focus on base 10 representation, arguing that memorability is subjective and base-dependent. Others appreciated the exploration of mathematical beauty and shared their own favorite "interesting" numbers. Several commenters noted the connection to Smarandache sequences and other recreational math concepts, with links provided for further exploration. The practicality of searching for such primes was also questioned, with some suggesting it was merely a curiosity with no real-world application.
Summary of Comments ( 71 )
https://news.ycombinator.com/item?id=43149883
HN commenters largely focused on the limitations and expansions of the puzzle. Some pointed out that the allowed operations weren't explicitly defined, leading to debates about the validity of certain solutions, particularly the use of the square root and floor/ceiling functions. Others discussed alternative approaches, such as using logarithms or the successor function. A few commenters explored variations of the puzzle, including using different numbers or a different quantity of the given number. The overall sentiment was one of intrigue, with many appreciating the puzzle's challenge and the creativity it sparked.
The Hacker News post titled "Making any integer with four 2s" has a modest number of comments, mostly focusing on variations of the puzzle and different mathematical approaches.
One commenter points out the ambiguity of the original problem statement regarding allowed operations. They clarify that the standard interpretation permits the use of basic arithmetic operations (addition, subtraction, multiplication, division), square roots, factorials, concatenation, decimal points, and sometimes powers and logs. They also suggest the use of the "binary concatenation operator", using .2 to represent 0.2 and creating arbitrary binary numbers, although acknowledging this might be considered "cheating."
Another commenter discusses the challenge of creating the number 7, mentioning strategies involving floor and ceiling functions applied to square roots and logs, ultimately leading to complex expressions. This commenter also highlights the importance of the specific allowed operations in determining the solvability of the puzzle for certain numbers.
The concept of the "four fours" puzzle is brought up, referencing a book ("Mathematical Recreations and Essays" by W. W. Rouse Ball) that discusses the puzzle and its variations. This commenter also suggests that the "four twos" version is likely more challenging.
Another comment thread discusses the importance of clearly defining the set of allowed operations, with one commenter specifically showing how to generate the numbers 1 through 10 using a relatively standard set of operations (including square root, factorial, concatenation, the floor function, and exponentiation).
Finally, one commenter introduces the concept of allowing the use of an infinite number of square roots, potentially simplifying the problem considerably, though deviating significantly from the typical rules.