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If (a, b, c) is a solution, then (ka, kb, kc) is also a solution for any k.Consequently, the solutions in rational numbers are all rescalings of integer solutions. Given an Euler brick with edge-lengths (a, b, c), the triple (bc, ac, ab) constitutes an Euler brick as well.
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 is a special case of the more general n queens problem of placing n non-attacking queens on an n×n chessboard.
Examples of broken diagonals from the number square in the image are as follows: 3,12,14,5; 10,1,7,16; 10,13,7,4; 15,8,2,9; 15,12,2,5; and 6,13,11,4. The fact that this square is a pandiagonal magic square can be verified by checking that all of its broken diagonals add up to the same constant:
The number of points (n), chords (c) and regions (r G) for first 6 terms of Moser's circle problem. In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem (named after Leo Moser), has a solution by an inductive method.
A magic square is an arrangement of numbers in a square grid so that the sum of the numbers along every row, column, and diagonal is the same. Similarly, one may define a magic cube to be an arrangement of numbers in a cubical grid so that the sum of the numbers on the four space diagonals must be the same as the sum of the numbers in each row, each column, and each pillar.
The grasshopper is a fairy chess piece that moves along ranks, files, and diagonals (like a queen) but only by hopping over another piece.The piece to be hopped may be of either color and any distance away, but the grasshopper must land on the square immediately beyond it in the same direction.
More generally, if the quadrilateral is a rectangle with sides a and b and diagonal d then Ptolemy's theorem reduces to the Pythagorean theorem. In this case the center of the circle coincides with the point of intersection of the diagonals. The product of the diagonals is then d 2, the right hand side of Ptolemy's relation is the sum a 2 + b 2.
In numerical linear algebra, the Jacobi method (a.k.a. the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations. Each diagonal element is solved for, and an approximate value is plugged in. The process is then iterated until it converges.