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The Pigeon Hole Principle: 7 gorgeous proofs

Mathologer・22 minutes read

"Récréation mathématique" is a 17th-century book filled with fun math problems, introducing the pigeonhole principle for the first time. The pigeonhole principle is a fundamental concept applied to various mathematical scenarios, offering simple yet profound solutions.

Insights

  • The "Récréation mathématique" book from the 17th century introduced the pigeonhole principle, a foundational proof technique stating that if there are more pigeons than pigeonholes, at least one hole will have more than one pigeon, showcasing its versatility in various scenarios like body hair counts, handshakes at parties, and Rubik's cube configurations.
  • The pigeonhole principle's application extends to diverse mathematical problems, from proving the existence of individuals with the same number of body hairs to demonstrating the inevitability of handshake twins at a party, emphasizing its simplicity yet profound solutions in resolving complex mathematical scenarios and encouraging exploration of its implications through engaging examples like the Rubik's cube and the Fitch Cheney five card trick.

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Recent questions

  • What is the pigeonhole principle?

    A simple concept stating if there are more pigeons than pigeonholes, at least one hole will have more than one pigeon. The principle is a powerful proof technique in mathematics.

  • How can fractions be expressed as decimals?

    Fractions can be expressed as decimals with infinite repeating tails. Every fraction has this property, such as the famous approximation of pi, 355/113, showcasing a repeating decimal pattern.

  • What is the handshake problem?

    At a party, there will always be at least two guests who shake hands with the exact same number of people. This is explained using scenarios with different numbers of guests, demonstrating the inevitability of finding handshake twins.

  • How can the Rubik's cube be solved?

    The Rubik's cube can be solved by repeating a single algorithm on a solved cube, eventually returning it to its solved state. The cube configurations represent pigeonholes, and repeating algorithms leads back to the solved cube.

  • What is the Fitch Cheney five card trick?

    The Fitch Cheney five card trick is a mathematical card trick named after mathematician William Fitch Cheney, Jr. It involves selecting 10 two-digit numbers to show that two collections with the same sum can be formed without overlapping numbers. It is considered one of the best mathematical card tricks ever invented.

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Summary

00:00

"Pigeonhole Principle: Math's Simple Power"

  • "Récréation mathématique" is a 17th-century book similar to Mathologer, filled with fun math problems created by Jesuit priests during their open days in 1626.
  • The book contains the first recorded instance of the pigeonhole principle, a simple yet powerful proof technique.
  • The pigeonhole principle states that if there are more pigeons than pigeonholes, at least one hole will have more than one pigeon.
  • Applying the principle to Australians with body hair, it's proven that there are individuals with the same number of body hairs.
  • The principle is further illustrated with examples involving pigeons on a sphere and the Rubik's cube.
  • Fractions can be expressed as decimals with infinite repeating tails, with every fraction having this property.
  • The fraction 355/113 is a famous approximation of pi, showcasing a repeating decimal pattern.
  • At a party, there will always be at least two guests who shake hands with the exact same number of people.
  • The handshake problem is explained using a scenario with seven guests, demonstrating the inevitability of finding handshake twins.
  • The pigeonhole principle is a fundamental concept that can be applied to various mathematical scenarios, providing simple yet profound solutions.

15:19

"Pigeon Party Theorem and Rubik's Cube"

  • The pigeonhole principle initially doesn't apply to a party with seven guests and seven possible handshake numbers.
  • If there is an unpopular guest (zero handshakes), there cannot be a super popular guest who shakes hands with everyone.
  • The possible handshake numbers at a party with seven guests are either zero to five or one to six.
  • The pigeonhole principle eventually works, showing that two guests must shake the same number of times.
  • The advanced pigeon party theorem states that at a party with six people, one of two scenarios will occur: three people shake hands with each other or three people do not shake hands with each other.
  • The Rubik's cube can be solved by repeating a single algorithm on a solved cube, eventually returning it to its solved state.
  • The process of solving the Rubik's cube is similar to fitting infinitely many pigeons into finitely many pigeonholes.
  • The Rubik's cube configurations represent the pigeonholes, and repeating algorithms eventually leads back to the solved cube.
  • The cycle length of a Rubik's cube algorithm cannot exceed 1260, regardless of complexity.
  • A mathematical olympiad problem involves selecting 10 two-digit numbers to show that two collections with the same sum can be formed without overlapping numbers.

30:38

Fitch Cheney Five Card Trick Ranking

  • The trick demonstrated is the Fitch Cheney five card trick, named after mathematician William Fitch Cheney, Jr. It is considered one of the best mathematical card tricks ever invented, and viewers are encouraged to rank their favorite proofs based on the pigeonhole principle in the comments.
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