A magic square is a square grid of numbers, such that the sum of each row, column, and diagonal is the same. In other words, the sum of any row, column, or diagonal of a magic square will always be the same constant value. Magic squares have been known and studied for thousands of years, and have fascinated mathematicians and scholars throughout history. In the world of computer programming, creating a magic square is a popular exercise that helps to develop algorithmic thinking and problem-solving skills. One common approach to creating a magic square in Java is by using a two-dimensional array. To create a magic square, we first need to determine the size of the square.
Throughout the article, we will examine the characteristics of Magic Numbers, discover their prevalence in mathematical puzzles and games, and explore their applications in programming scenarios. We will also delve into techniques for identifying and utilizing Magic Numbers efficiently in Java.
Magic Square of size 3 ----------------------- 2 7 6 9 5 1 4 3 8 Sum in each row each column 3 3 2 1 2 15 Magic Square of size 5 ---------------------- 9 3 22 16 15 2 21 20 14 8 25 19 13 7 1 18 12 6 5 24 11 10 4 23 17 Sum in each row each column 5 5 2 1 2 65 Magic Square of size 7 ---------------------- 20 12 4 45 37 29 28 11 3 44 36 35 27 19 2 43 42 34 26 18 10 49 41 33 25 17 9 1 40 32 24 16 8 7 48 31 23 15 14 6 47 39 22 21 13 5 46 38 30 Sum in each row each column 7 7 2 1 2 175. If the magic square already contains a number at the calculated position, calculated column position will be decremented by 2, and calculated row position will be incremented by 1.
To create a magic square, we first need to determine the size of the square. The size of a magic square is always an odd number, such as 3, 5, 7, and so on. Let's assume we want to create a 3x3 magic square.
MagicSquare.java
Below is the syntax highlighted version of MagicSquare.java from §1.4 Arrays.
/****************************************************************************** * Compilation: javac MagicSquare.java * Execution: java MagicSquare n * * Generates a magic square of order n. A magic squares is an n-by-n * matrix of the integers 1 to n^2, such that all row, column, and * diagonal sums are equal. * * One way to generate a magic square when n is odd is to assign * the integers 1 to n^2 in ascending order, starting at the * bottom, middle cell. Repeatedly assign the next integer to the * cell adjacent diagonally to the right and down. If this cell * has already been assigned another integer, instead use the * cell adjacently above. Use wrap-around to handle border cases. * * * % java MagicSquare 3 * 4 9 2 * 3 5 7 * 8 1 6 * * % java MagicSquare 5 * 11 18 25 2 9 * 10 12 19 21 3 * 4 6 13 20 22 * 23 5 7 14 16 * 17 24 1 8 15 * * Limitations * ----------- * - n must be odd * ******************************************************************************/ public class MagicSquare public static void main(String[] args) int n = Integer.parseInt(args[0]); if (n % 2 == 0) throw new RuntimeException("n must be odd"); int[][] magic = new int[n][n]; int row = n-1; int col = n/2; magic[row][col] = 1; for (int i = 2; i n*n; i++) if (magic[(row + 1) % n][(col + 1) % n] == 0) row = (row + 1) % n; col = (col + 1) % n; > else row = (row - 1 + n) % n; // don't change col > magic[row][col] = i; > // print results for (int i = 0; i n; i++) for (int j = 0; j n; j++) if (magic[i][j] 10) System.out.print(" "); // for alignment if (magic[i][j] 100) System.out.print(" "); // for alignment System.out.print(magic[i][j] + " "); > System.out.println(); > > >
Next, we can initialize a two-dimensional array of size 3x3, which will represent our magic square. We can start by filling the array with zeros. To fill the magic square, we need to place the numbers 1 to 9 in a specific pattern. The pattern for a 3x3 magic square is as follows: 1 2 3 4 5 6 7 8 9 We can achieve this pattern by assigning the numbers in a clockwise spiral manner, starting from the center of the square. We can use a nested loop to iterate over the rows and columns of the array, and update the values accordingly. After filling the magic square, we can print the array to see the result. The sum of any row, column, or diagonal should be the same constant value. Creating a magic square in Java is not only a fun exercise, but it also helps to develop skills such as problem-solving, algorithmic thinking, and pattern recognition. It is a great way to explore the world of programming while also exploring the fascinating world of mathematics..
Reviews for "Introducing Magic Cubes: 3D Extensions of Magic Squares in Java"
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