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27-695: LXV may refer to: 65 (number) in Roman numerals LXV, the IATA and FAA LID code for Lake County Airport , United States Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title LXV . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=LXV&oldid=1109717755 " Category : Disambiguation pages Hidden categories: Short description

54-500: A local optimum. (In such a case, the algorithm may be restarted with a different initial configuration.) On the other hand, it can solve problem sizes that are several orders of magnitude beyond the scope of a depth-first search. As an alternative to backtracking, solutions can be counted by recursively enumerating valid partial solutions, one row at a time. Rather than constructing entire board positions, blocked diagonals and columns are tracked with bitwise operations. This does not allow

81-487: A set of six objects into four non-empty subsets. 65 = 1 + 2 + 3 + 4 + 5 . 65 is the length of the hypotenuse of 4 different Pythagorean triangles , the lowest number to have more than 2: 65 = 16 + 63 = 33 + 56 = 39 + 52 = 25 + 60 . The first two are "primitive", and 65 is the lowest number to be the largest side of more than one such triple. 65 is the number of compositions of 11 into distinct parts. Eight queens puzzle The eight queens puzzle

108-440: A simple rule that chooses one queen from each column, it is possible to reduce the number of possibilities to 16,777,216 (that is, 8 ) possible combinations. Generating permutations further reduces the possibilities to just 40,320 (that is, 8! ), which can then be checked for diagonal attacks. The eight queens puzzle has 92 distinct solutions. If solutions that differ only by the symmetry operations of rotation and reflection of

135-450: A way that is much more efficient than the naïve brute-force search algorithm, which considers all 64  = 2  = 281,474,976,710,656 possible blind placements of eight queens, and then filters these to remove all placements that place two queens either on the same square (leaving only 64!/56! = 178,462,987,637,760 possible placements) or in mutually attacking positions. This very poor algorithm will, among other things, produce

162-499: Is a constant that lies between 1.939 and 1.945. (Here o (1) represents little o notation .) If one instead considers a toroidal chessboard (where diagonals "wrap around" from the top edge to the bottom and from the left edge to the right), it is only possible to place n queens on an n × n {\displaystyle n\times n} board if n ≡ 1 , 5 mod 6. {\displaystyle n\equiv 1,5\mod 6.} In this case,

189-407: Is a special case of the more general n queens problem of placing n non-attacking queens on an n × n chessboard. Solutions exist for all natural numbers n with the exception of n = 2 and n = 3. Although the exact number of solutions is only known for n ≤ 27, the asymptotic growth rate of the number of solutions is approximately (0.143 n ) . Chess composer Max Bezzel published

216-448: Is also the magic constant of n-Queens Problem for n = 5. 65 is the smallest integer that can be expressed as a sum of two distinct positive squares in two (or more) ways, 65 = 8 + 1 = 7 + 4 . It appears in the Padovan sequence , preceded by the terms 28, 37, 49 (it is the sum of the first two of these). 65 is a Stirling number of the second kind , the number of ways of dividing

243-747: Is an octagonal number . It is also a Cullen number . Given 65, the Mertens function returns 0 . This number is the magic constant of a 5x5 normal magic square : [ 17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9 ] . {\displaystyle {\begin{bmatrix}17&24&1&8&15\\23&5&7&14&16\\4&6&13&20&22\\10&12&19&21&3\\11&18&25&2&9\end{bmatrix}}.} This number

270-411: Is different from Wikidata All article disambiguation pages All disambiguation pages 65 (number) 65 is the nineteenth distinct semiprime , (5.13); and the third of the form (5.q), where q is a higher prime. 65 has a prime aliquot sum of 19 within an aliquot sequence of one composite numbers (65, 19 , 1 ,0) to the prime; as the first member' of the 19 -aliquot tree. It

297-401: Is identical to its own 180° rotation, so has only four variants (itself and its reflection, its 90° rotation and the reflection of that). Thus, the total number of distinct solutions is 11×8 + 1×4 = 92. All fundamental solutions are presented below: Solution 10 has the additional property that no three queens are in a straight line . Brute-force algorithms to count

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324-508: Is the highest-order board that has been completely enumerated. The following tables give the number of solutions to the n queens problem, both fundamental (sequence A002562 in the OEIS ) and all (sequence A000170 in the OEIS ), for all known cases. The number of placements in which furthermore no three queens lie on any straight line is known for n ≤ 25 {\displaystyle n\leq 25} (sequence A365437 in

351-411: Is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; thus, a solution requires that no two queens share the same row, column, or diagonal. There are 92 solutions. The problem was first posed in the mid-19th century. In the modern era, it is often used as an example problem for various computer programming techniques. The eight queens puzzle

378-418: Is used as an example of a problem that can be solved with a recursive algorithm , by phrasing the n queens problem inductively in terms of adding a single queen to any solution to the problem of placing n −1 queens on an n × n chessboard. The induction bottoms out with the solution to the 'problem' of placing 0 queens on the chessboard, which is the empty chessboard. This technique can be used in

405-674: The OEIS ). In 2021, Michael Simkin proved that for large numbers n , the number of solutions of the n queens problem is approximately ( 0.143 n ) n {\displaystyle (0.143n)^{n}} . More precisely, the number Q ( n ) {\displaystyle {\mathcal {Q}}(n)} of solutions has asymptotic growth Q ( n ) = ( ( 1 ± o ( 1 ) ) n e − α ) n {\displaystyle {\mathcal {Q}}(n)=((1\pm o(1))ne^{-\alpha })^{n}} where α {\displaystyle \alpha }

432-561: The asymptotic number of solutions is T ( n ) = ( ( 1 + o ( 1 ) ) n e − 3 ) n . {\displaystyle T(n)=((1+o(1))ne^{-3})^{n}.} Finding all solutions to the eight queens puzzle is a good example of a simple but nontrivial problem. For this reason, it is often used as an example problem for various programming techniques, including nontraditional approaches such as constraint programming , logic programming or genetic algorithms . Most often, it

459-417: The board are counted as one, the puzzle has 12 solutions. These are called fundamental solutions; representatives of each are shown below. A fundamental solution usually has eight variants (including its original form) obtained by rotating 90, 180, or 270° and then reflecting each of the four rotational variants in a mirror in a fixed position. However, one of the 12 fundamental solutions (solution 12 below)

486-431: The board at a time, eliminating most nonsolution board positions at a very early stage in their construction. Because it rejects rook and diagonal attacks even on incomplete boards, it examines only 15,720 possible queen placements. A further improvement, which examines only 5,508 possible queen placements, is to combine the permutation based method with the early pruning method: the permutations are generated depth-first, and

513-417: 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. The backtracking depth-first search program, a slight improvement on the permutation method, constructs the search tree by considering one row of

540-539: The eight queens puzzle in 1848. Franz Nauck published the first solutions in 1850. Nauck also extended the puzzle to the n queens problem, with n queens on a chessboard of n × n squares. Since then, many mathematicians , including Carl Friedrich Gauss , have worked on both the eight queens puzzle and its generalized n -queens version. In 1874, S. Günther proposed a method using determinants to find solutions. J.W.L. Glaisher refined Gunther's approach. In 1972, Edsger Dijkstra used this problem to illustrate

567-453: The following formulas. Let ( i , j ) be the square in column i and row j on the n × n chessboard, k an integer. One approach is For n  = 8 this results in fundamental solution 1 above. A few more examples follow. There is no known formula for the exact number of solutions for placing n queens on an n × n board i.e. the number of independent sets of size n in an n × n queen's graph . The 27×27 board

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594-431: The number of solutions are computationally manageable for n  = 8 , but would be intractable for problems of n  ≥ 20 , as 20! = 2.433 × 10 . If the goal is to find a single solution, one can show solutions exist for all n ≥ 4 with no search whatsoever. These solutions exhibit stair-stepped patterns, as in the following examples for n = 8, 9 and 10: The examples above can be obtained with

621-440: The placement of the queens. The ' minimum-conflicts ' heuristic – moving the piece with the largest number of conflicts to the square in the same column where the number of conflicts is smallest – is particularly effective: it easily finds a solution to even the 1,000,000 queens problem. Unlike the backtracking search outlined above, iterative repair does not guarantee a solution: like all greedy procedures, it may get stuck on

648-517: The power of what he called structured programming . He published a highly detailed description of a depth-first backtracking algorithm . The problem of finding all solutions to the 8-queens problem can be quite computationally expensive, as there are 4,426,165,368 possible arrangements of eight queens on an 8×8 board, but only 92 solutions. It is possible to use shortcuts that reduce computational requirements or rules of thumb that avoids brute-force computational techniques . For example, by applying

675-472: The recovery of individual solutions. The following program is a translation of Niklaus Wirth 's solution into the Python programming language, but does without the index arithmetic found in the original and instead uses lists to keep the program code as simple as possible. By using a coroutine in the form of a generator function , both versions of the original can be unified to compute either one or all of

702-423: The same results over and over again in all the different permutations of the assignments of the eight queens, as well as repeating the same computations over and over again for the different sub-sets of each solution. A better brute-force algorithm places a single queen on each row, leading to only 8  = 2  = 16,777,216 blind placements. It is possible to do much better than this. One algorithm solves

729-418: The search space is pruned if the partial permutation produces a diagonal attack. Constraint programming can also be very effective on this problem. An alternative to exhaustive search is an 'iterative repair' algorithm, which typically starts with all queens on the board, for example with one queen per column. It then counts the number of conflicts (attacks), and uses a heuristic to determine how to improve

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