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  2. Collatz conjecture - Wikipedia

    en.wikipedia.org/wiki/Collatz_conjecture

    The Collatz conjecture is: This process will eventually reach the number 1, regardless of which positive integer is chosen initially. That is, for each , there is some with . If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1.

  3. Continued fraction - Wikipedia

    en.wikipedia.org/wiki/Continued_fraction

    The continued fraction representation for a real number is finite if and only if it is a rational number. In contrast, the decimal representation of a rational number may be finite, for example ⁠ 137 / 1600 ⁠ = 0.085625, or infinite with a repeating cycle, for example ⁠ 4 / 27 ⁠ = 0.148148148148...

  4. Egyptian fraction - Wikipedia

    en.wikipedia.org/wiki/Egyptian_fraction

    Many of the calculations throughout Fibonacci's book involve numbers represented as Egyptian fractions, and one section of this book [11] provides a list of methods for conversion of vulgar fractions to Egyptian fractions. If the number is not already a unit fraction, the first method in this list is to attempt to split the numerator into a sum ...

  5. Bertrand's box paradox - Wikipedia

    en.wikipedia.org/wiki/Bertrand's_box_paradox

    Bertrand's box paradox is a veridical paradox in elementary probability theory. It was first posed by Joseph Bertrand in his 1889 work Calcul des Probabilités. There are three boxes: a box containing two gold coins, a box containing two silver coins, a box containing one gold coin and one silver coin. A coin withdrawn at random from the three ...

  6. Middle-square method - Wikipedia

    en.wikipedia.org/wiki/Middle-square_method

    Directed graph of all 100 2-digit pseudorandom numbers obtained using the middle-square method with n = 2. In mathematics and computer science, the middle-square method is a method of generating pseudorandom numbers. In practice it is a highly flawed method for many practical purposes, since its period is usually very short and it has some ...

  7. Four fours - Wikipedia

    en.wikipedia.org/wiki/Four_fours

    Four fours. Four fours is a mathematical puzzle, the goal of which is to find the simplest mathematical expression for every whole number from 0 to some maximum, using only common mathematical symbols and the digit four. No other digit is allowed. Most versions of the puzzle require that each expression have exactly four fours, but some ...

  8. Euclidean algorithm - Wikipedia

    en.wikipedia.org/wiki/Euclidean_algorithm

    The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since ...

  9. Bertrand's ballot theorem - Wikipedia

    en.wikipedia.org/wiki/Bertrand's_ballot_theorem

    The variant problem can be solved by the reflection method in a similar way to the original problem. The number of possible vote sequences is (+). Call a sequence "bad" if the second candidate is ever ahead, and if the number of bad sequences can be enumerated then the number of "good" sequences can be found by subtraction and the probability ...

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