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How did Ramanujan solve the STRAND puzzle?

Mathologer・2 minutes read

Srinivasa Ramanujan effortlessly solves complex mathematical puzzles using infinite fractions, showcasing his genius in relating solutions to Pell equations and offering efficient problem-solving methods. Ramanujan's insights into continued fractions, like those for root 2, demonstrate the connection to the Euclidean algorithm, proving the irrationality of root 2 and providing a visual representation for understanding and calculating continued fractions.

Insights

  • Ramanujan's ability to effortlessly solve complex mathematical puzzles using infinite fractions showcased his exceptional mathematical brilliance and deep understanding of equations, as seen in his solution to The Strand magazine puzzle and the related Pell equation.
  • The connection between continued fractions, like the one Ramanujan used, and the Euclidean algorithm not only sheds light on the nature of these mathematical tools but also proves the irrationality of root 2, emphasizing the intricate relationships between different mathematical concepts and their applications in solving problems efficiently.

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  • Who was Srinivasa Ramanujan?

    A mathematical genius known for complex equations.

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Summary

00:00

Ramanujan's Infinite Fraction Solves Complex Puzzles

  • Srinivasa Ramanujan, an Indian mathematical genius, is known for his complex infinite continued fraction equations involving pi, phi, and e.
  • Ramanujan's friend, Prasantha Mahalanobis, visited him in Cambridge in 1914, where they encountered a puzzle from The Strand magazine involving a man living on a street with equal sums of house numbers on each side.
  • The puzzle required finding the number of houses on the street and the man's house number, with Mahalanobis eventually solving it and challenging Ramanujan.
  • Ramanujan effortlessly provided the solution using an infinite fraction, which not only solved The Strand puzzle but also related to a Pell equation.
  • Dudeney's puzzle specified more than 50 and less than 500 houses on the street, leading to multiple solutions beyond the initial one found.
  • Ramanujan's infinite fraction method revealed all solutions at once, showcasing his mathematical brilliance.
  • Algebraic manipulations led to an equation with x and y representing the number of houses and the special house number, respectively.
  • Ramanujan recognized that all solutions to the equation were contained in a special infinite fraction, which he immediately grasped.
  • Calculating partial fractions from the infinite fraction extracted solutions to the equation, with every second fraction corresponding to positive integer solutions.
  • Translating these fractions provided solutions to the original puzzle, showcasing the power of Ramanujan's method in solving complex mathematical problems efficiently.

15:49

Ramanujan's Infinite Fractions and Mathematical Insights

  • Ramanujan's mathematical insights were impressive, particularly his understanding of infinite fractions.
  • Continued fractions, like the one Ramanujan used, were familiar to mathematicians a century ago.
  • Ramanujan's solution, while remarkable, was not as unique as commonly believed due to the historical context of continued fractions.
  • The equation Ramanujan solved is a well-known Pell equation, with solutions found thousands of years ago.
  • The solutions to the Pell equation are contained within the simple continued infinite fraction of root k.
  • The connection between the infinite fraction and root 2 is explained through the Euclidean algorithm.
  • The Euclidean algorithm, a mathematical tool capturing the highest common factor of two numbers, is linked to continued fractions like the one for root 2.
  • The Euclidean algorithm's visual representation involves chopping off squares from a rectangle to find the highest common factor.
  • The continued fraction for root 2 is derived from the Euclidean algorithm, proving root 2 is irrational.
  • Understanding continued fractions through visual representations provides insight into their nature and calculation methods.

32:45

"Approximating Root 2 with Partial Fractions"

  • The value 17 over 12 is a good approximation of root two, with the infinite fraction getting closer to root 2 the further down we go.
  • Continued fractions are discussed in this video, with partial fractions being the best rational approximations to full fractions.
  • The difference between a fraction and root 2 is a measure of how good the approximation is, with partial fractions in lowest terms scoring one.
  • Partial fractions are best in a simpler sense as they are closest to root 2 among all fractions with denominators less than a certain number.
  • A method to generate all partial fractions involves removing the two smallest squares and smoothing out the indents.
  • Another method involves removing the largest two squares to construct early partial fractions, with a simple relationship between successive partial fractions.
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