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.
This blog post, titled "Making any integer with four 2s," delves into the mathematical puzzle of representing any integer using exactly four instances of the digit 2 and a limited set of operations. The author meticulously explores various mathematical tools and techniques to achieve this goal. Starting with the basic arithmetic operations of addition, subtraction, multiplication, and division, the author demonstrates how combinations of these operations, along with the use of parentheses for enforcing order of operations, can generate a surprising number of integers.
The author then introduces the concept of concatenation, which allows combining two twos to form 22, significantly expanding the range of achievable numbers. Further expanding the toolkit, the author incorporates the use of the square root, factorial, and decimal point (to create 2.2 or .2), illustrating how these additions further broaden the scope of constructible integers. For instance, the factorial operation allows for the creation of larger numbers like 22! or combinations like (2+2)!.
The author acknowledges the potential ambiguity in interpreting concatenation and the decimal point and clarifies their intended usage within this puzzle's framework. To represent powers, the author employs the convention of repeated square roots, recognizing this as a slightly unconventional but effective representation within the constraints of the four-twos rule.
The exploration goes beyond basic arithmetic operations, introducing the concept of logarithms, specifically base 10 and base 2 logarithms, represented as lg and lb respectively. The author carefully explains how logarithms can be used strategically in conjunction with the other allowed operations to generate specific integers, showcasing the power and versatility this mathematical function brings to the problem.
The post demonstrates the process of constructing several specific integers as examples, highlighting the creativity and logical thinking required to manipulate the four twos into the desired result. It emphasizes the intricate interplay of different mathematical operations and how their clever combination can yield a wide range of integer values. The author concludes by inviting readers to explore and discover further possibilities within the established rules, encouraging continued engagement with this mathematical challenge.
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.