The Archimedes Number - Numberphile
TLDRThe video script narrates the intriguing tale of Archimedes' Cattle Problem, once the most famous unsolved math problem. Discovered in a manuscript by Gotthold Ephraim Lessing in 1773, the problem is presented as a poem in ancient Greek, describing a complex relationship between different colored cattle. It challenges solvers to find a number so vast that if each represented an atom, it would exceed the number of atoms in the known universe. The script takes viewers through the history of attempts to solve the problem, highlighting the efforts of mathematicians like Gauss and the eventual use of supercomputers in 1965 to fully calculate the immense number.
Takeaways
- 📜 Archimedes' cattle problem was once the most famous unsolved problem in mathematics but is now relatively unknown.
- 🎭 The problem was discovered in 1773 by Gotthold Ephraim Lessing, a German intellectual and librarian, in a manuscript at Wolfenbüttel.
- 📝 The manuscript contained a poem in Greek, a letter from Archimedes to Eratosthenes, describing the cattle problem with twenty-two rhyming couplets.
- 🐂 The problem involves a herd of cattle with various colors and relationships described by fractions, leading to a square and a triangular number.
- 🤔 The problem was initially considered dull and difficult, but it later became a celebrated challenge in mathematics.
- 🔢 The solution to the first part of the problem, involving the number of cattle, was found to be over fifty million, which was a surprising result.
- 🌍 The second part of the problem, involving the sum of dappled and yellow bulls as a triangular number, was much more complex and remained unsolved for a long time.
- 🧐 The problem was rumored to have been solved by Gauss but his solution was never seen, and it wasn't until 1965 that the full number was computed with the help of supercomputers.
- 📈 The final number was incredibly large, beginning with '766' and continuing for over 206,542 digits, a number so large it would exceed the universe if each represented an atom.
- 🤷♂️ There is debate over whether the problem was actually authored by Archimedes, but his interest in large numbers and his work in The Sand Reckoner suggest a connection.
- 📚 The video recommends Alex Bellos' book 'So You Think You've Got Problems?' for more puzzles and teasers like the cattle problem.
Q & A
Who is Archimedes and why is he significant in the context of the cattle problem?
-Archimedes is an ancient Greek mathematician, physicist, and inventor, renowned as one of the most brilliant minds in history. In the context of the cattle problem, he is attributed with the creation of this complex and famous mathematical puzzle.
What is the significance of the year 1773 in relation to the cattle problem?
-In 1773, the German intellectual Gotthold Ephraim Lessing discovered a manuscript containing the cattle problem, which was previously unknown. This marked the rediscovery of the problem and its introduction to the modern mathematical community.
What was the medium in which the cattle problem was originally presented?
-The cattle problem was originally presented as a poem with twenty-two rhyming couplets in ancient Greek, found in a manuscript written by Archimedes to Eratosthenes.
What is the basic premise of the cattle problem?
-The cattle problem involves a hypothetical scenario where the Sun-God has a herd of cattle of various colors on the island of Sicily. The problem presents a series of relationships between the numbers of different colored bulls and cows, which are described using fractions and leads to the identification of a square and a triangular number.
What are square numbers and triangular numbers as mentioned in the cattle problem?
-Square numbers are integers that are the square of another integer (e.g., 1, 4, 9, 16, 25, etc.). Triangular numbers are numbers that can form an equilateral triangle when arranged in dots (e.g., 1, 3, 6, 10, etc.).
Why is the cattle problem considered 'boring' by the narrator?
-The narrator finds the cattle problem boring due to its complexity and the seemingly mundane nature of the relationships between the numbers of different colored cattle, which may not initially appear to be as engaging as other mathematical problems.
What was the initial solution to the cattle problem when it was first rediscovered?
-The initial solution to the cattle problem, when it was first rediscovered, estimated the total number of cattle to be just over fifty million.
How did the solution to the cattle problem evolve over time?
-After the initial solution, further analysis revealed that the number of cattle in the herd was actually 51 trillion, which was a staggering increase. Later, it was found that including the condition of the dappled bulls plus yellow bulls being a triangular number made the problem unsolvable with 18th-century mathematics.
Who was August Amthor and what was his contribution to solving the cattle problem?
-August Amthor was a German mathematician who, about a hundred years after the problem was discovered, made progress by identifying that the solution began with the digits '766' and continued for another 206,542 digits.
What role did supercomputers play in the final solution of the cattle problem?
-Supercomputers in 1965 were used to calculate the full number of the cattle problem, which took about seven hours and resulted in a printout that stretched to 42 sheets of A4 paper.
What is the debate surrounding the authorship of the cattle problem?
-There is a debate among scholars about whether the cattle problem was indeed authored by Archimedes. While the problem states it is from Archimedes to Eratosthenes, some researchers question its authenticity, while others believe it aligns with Archimedes' known interest in large numbers.
How did the cattle problem demonstrate the power of mathematics according to the script?
-The cattle problem demonstrates the power of mathematics by showing that with just nine simple statements, one can create a problem that describes an unimaginably large number, which took over 2,000 years to solve, reflecting the genius of mathematical problem-solving.
Outlines
📜 Archimedes' Cattle Problem Introduction
This paragraph introduces the Archimedes' cattle problem, a once famous but now obscure mathematical challenge. Archimedes, renowned as an ancient mathematician, is the attributed author of this problem. The problem was discovered in 1773 by Gotthold Ephraim Lessing in a Greek manuscript at the library of Wolfenbüttel. The manuscript contained a letter from Archimedes to Eratosthenes, written as a poem with 22 rhyming couplets, which described a complex problem involving a herd of cattle with various colors and relationships. The problem is presented in modern language, involving fractions and relationships between the numbers of different colored bulls and cows, leading to a square and a triangular number. The paragraph humorously acknowledges the problem's complexity and dullness while setting the stage for a fascinating historical narrative.
🐄 The Enormity of the Cattle Problem
This paragraph delves into the specifics and the historical attempts to solve the Archimedes' cattle problem. The problem, despite its initial straightforward appearance, quickly escalates in complexity when considering the additional conditions that the sum of white and black bulls forms a square number, and the sum of dappled and yellow bulls forms a triangular number. The first solution, found shortly after the problem's discovery, suggested a herd size of over fifty million cattle. Later, the problem was further solved to reveal a herd size of 51 trillion, an astronomical number that, if taken literally, would cover the entire island of Sicily in cattle. The paragraph also touches on the historical significance of the problem, its difficulty, and the involvement of renowned mathematicians like Gauss. It culminates in the realization that the full solution to the problem, involving an enormous number with digits beginning '766' and continuing for over 200,000 digits, was only fully computed with the advent of supercomputers in the 20th century. The paragraph concludes with a reflection on the authorship and the possible intent behind the problem, suggesting it may have been a demonstration of mathematical prowess rather than a solvable problem.
Mindmap
Keywords
💡Archimedes
💡Cattle Problem
💡Gotthold Ephraim Lessing
💡Eratosthenes
💡Rhyming Couplets
💡Square Number
💡Triangular Number
💡Simultaneous Equations
💡Carl Friedrich Gauss
💡August Amthor
💡Supercomputers
Highlights
Archimedes' cattle problem was once the most famous unsolved problem in mathematics.
The problem was discovered in 1773 by Gotthold Ephraim Lessing in a Greek manuscript.
The manuscript contained a letter from Archimedes to Eratosthenes in the form of a poem with 22 rhyming couplets.
The problem involves a herd of cattle with specific ratios of different colored bulls and cows.
The problem was initially considered boring and dull, but it later revealed a fascinating mathematical challenge.
The cattle problem involves unit fractions and the concept of square and triangular numbers.
The initial solution to the problem suggested a herd of just over fifty million cattle.
The problem was later solved to reveal a number in the trillions, indicating a herd size of 51 trillion.
The final part of the problem, involving dappled and yellow bulls, was beyond 18th-century mathematics.
Gauss, a renowned mathematician, was rumored to have a proof for the problem but it was never seen.
August Amthor, a German mathematician, found the beginning digits of the solution but not the full answer.
The full solution to the problem was not achieved until 1965 with the help of supercomputers.
The complete solution was a number with 206,542 digits, making it an enormous and incomprehensible figure.
The problem may have been created by Archimedes to demonstrate the power of mathematics rather than to be solved.
Archimedes was known for his interest in massive numbers, as shown in 'The Sand Reckoner'.
The problem's authorship is debated, with some believing it to be from Archimedes due to his fascination with large numbers.
The cattle problem showcases the ability of simple mathematical statements to describe an unimaginably large number.
The video recommends Alex Bellos' book 'So You Think You've Got Problems?' for more puzzles and teasers.
Transcripts
Browse More Related Video
5.0 / 5 (0 votes)
Thanks for rating: