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.
Calcverse is a collection of simple, focused online calculators built by a solo developer as a counterpoint to the current hype around AI agents. The creator emphasizes the value of small, well-executed tools that solve specific problems efficiently. The calculators currently offered on the site cover areas like loan comparisons, unit conversions, and investment calculations, with more planned for the future. The project embraces a minimalist design and aims to provide a practical alternative to overly complex software.
HN users generally praised the calculator's clean UI/UX and appreciated the developer's focus on a simple, well-executed project rather than chasing the AI hype. Several commenters suggested potential improvements or expansions, including adding more unit conversions, financial calculators, and even integrating with existing tools like Excel or Google Sheets. Some pointed out the existing prevalence of specialized online calculators, questioning the project's long-term viability. Others expressed interest in the technical implementation details, particularly the use of Qwik and Partytown. A few jokingly questioned the project's description as "just" calculators, recognizing the complexity and value in building a robust and user-friendly calculation tool.
The blog post explores the surprising observation that repeated integer addition can approximate floating-point multiplication, specifically focusing on the case of multiplying by small floating-point numbers slightly greater than one. It explains this phenomenon by demonstrating how the accumulation of fractional parts during repeated addition mimics the effect of multiplication. When adding a floating-point number slightly larger than one to itself repeatedly, the fractional part grows with each addition, eventually getting large enough to increment the integer part. This stepping increase in the integer part, combined with the accumulating fractional component, closely resembles the scaling effect of multiplication by that same number. The post illustrates this relationship using both visual representations and mathematical explanations, linking the behavior to the inherent properties of floating-point numbers and their representation in binary.
Hacker News commenters generally praised the article for clearly explaining a non-obvious relationship between integer addition and floating-point multiplication. Some highlighted the practical implications, particularly in older hardware or specialized situations where integer operations are significantly faster. One commenter pointed out the historical relevance to Quake III's fast inverse square root approximation, while another noted the connection to logarithms and how this technique could be extended to other operations. A few users discussed the limitations and boundary conditions, emphasizing the approximation's validity only within specific ranges and the importance of understanding those constraints. Some commenters provided further context by linking to related concepts like the "magic number" used in the Quake III algorithm and resources on floating-point representation.
This paper investigates how pre-trained large language models (LLMs) perform integer addition. It finds that LLMs, despite lacking explicit training on arithmetic, learn to leverage positional encoding based on Fourier features to represent numbers internally. This allows them to achieve surprisingly good accuracy on addition tasks, particularly within the range of numbers present in their training data. The authors demonstrate this by analyzing attention patterns and comparing LLM performance with models using alternative positional encodings. They also show how manipulating or ablating these Fourier features directly impacts the models' ability to add, strongly suggesting that LLMs have implicitly learned a form of Fourier-based arithmetic.
Hacker News users discussed the surprising finding that LLMs appear to use Fourier features internally to perform addition, as indicated by the linked paper. Several commenters expressed fascination with this emergent behavior, highlighting how LLMs discover and utilize mathematical concepts without explicit instruction. Some questioned the paper's methodology and the strength of its conclusions, suggesting alternative explanations or calling for further research to solidify the claims. A few users also discussed the broader implications of this discovery for understanding how LLMs function and how they might be improved. The potential link to the Fourier-based positional encoding used in Transformer models was also noted as a possible contributing factor.
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.