Calculating the Number of Order-6 Magic Squares with Modular Lifting
نویسندگان
چکیده
Introduction There is no known general formula to calculate the total number of magic squares of a given order. However, using various methods, the number of magic squares has been calculated for orders smaller than 6. In this work we describe an approach to bounding the computational complexity of enumerating order-6 squares and provide a related method to approximate the number of such squares. Though many similar definitions of magic squares appear in the literature, we restrict ourselves here to the “traditional” definition. A magic square of order n is an n × n array containing an arrangement of each of the numbers 1 through n such that the numbers in every column, row, and diagonal sum to the same number. This number M is called the magic constant, which can be expressed as:
منابع مشابه
Inverse modeling of gravity field data due to finite vertical cylinder using modular neural network and least-squares standard deviation method
In this paper, modular neural network (MNN) inversion has been applied for the parameters approximation of the gravity anomaly causative target. The trained neural network is used for estimating the amplitude coefficient and depths to the top and bottom of a finite vertical cylinder source. The results of the applied neural network method are compared with the results of the least-squares stand...
متن کاملLecture 5 : Latin Squares and Magic
Today's application is to magic! Not the friendship kind, though 1 ; instead, we're going to talk about magic squares, an incredibly old piece of mathematics that we can study using Latin squares. Definition. A magic square is a n × n grid filled with the integers {0, 1,. .. n 2 − 1}, such that • each number is used exactly once in our entire grid, and • the sum of all of the entries along any ...
متن کاملA ug 2 00 9 ENUMERATION OF 4 × 4 MAGIC SQUARES
A magic square is an n × n array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for n = 4, as a function of either the magic sum or an upper bound on the entries. The previous record for both functions was the n = 3 case. Our methods are based on inside-out polytopes, i.e., the combination of hyperplane a...
متن کاملJ an 2 00 5 The Number of “ Magic ” Squares , Cubes , and Hypercubes 1
Magic squares have turned up throughout history, some in a mathematical context, others in philosophical or religious contexts. According to legend, the first magic square was discovered in China by an unknown mathematician sometime before the first century A.D. It was a magic square of order three thought to have appeared on the back of a turtle emerging from a river. Other magic squares surfa...
متن کاملConstructing all magic squares of order three
A magic square of order n is an n by n matrix with distinct nonnegative integer entries such that every row sum, column sum, and (two) diagonal sums equals to the same number m, the magic number. Adding 1 to every entry will give us a traditional magic square of positive integers. A magic square is pure if the entries are the consecutive numbers from 0 to n − 1, and hence it has magic number 3 ...
متن کامل