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