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In computer science and mathematics, the Josephus problem (or Josephus permutation) is a theoretical problem related to a certain counting-out game. Such games are used to pick out a person from a group, e.g. eeny, meeny, miny, moe. A drawing for the Josephus problem sequence for 500 people and skipping value of 6.
This would have been the first attempt on record to solve a difficult problem in permutations and combinations. [4] Al-Khalil (717–786), an Arab mathematician and cryptographer, wrote the Book of Cryptographic Messages. It contains the first use of permutations and combinations, to list all possible Arabic words with and without vowels. [5]
The ! permutations of the numbers from 1 to may be placed in one-to-one correspondence with the ! numbers from 0 to ! by pairing each permutation with the sequence of numbers that count the number of positions in the permutation that are to the right of value and that contain a value less than (that is, the number of inversions for which is the ...
Any Hamiltonian path through the created graph is a superpermutation, and the problem of finding the path with the smallest weight becomes a form of the traveling salesman problem. The first instance of a superpermutation smaller than length 1 ! + 2 ! + … + n ! {\displaystyle 1!+2!+\ldots +n!} was found using a computer search on this method ...
Since factors as (+ +) (+ +) in [], the group G contains a permutation that is a product of disjoint cycles of lengths 2 and 3 (in general, when a monic integer polynomial reduces modulo a prime to a product of distinct monic irreducible polynomials, the degrees of the factors give the lengths of the disjoint cycles in some permutation ...
In a 1977 review of permutation-generating algorithms, Robert Sedgewick concluded that it was at that time the most effective algorithm for generating permutations by computer. [2] The sequence of permutations of n objects generated by Heap's algorithm is the beginning of the sequence of permutations of n+1 objects.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
One algorithm solves the eight rooks puzzle by generating the permutations of the numbers 1 through 8 (of which there are 8! = 40,320), and uses the elements of each permutation as indices to place a queen on each row. Then it rejects those boards with diagonal attacking positions. This animation illustrates backtracking to solve the problem. A ...