How to Solve Magic Squares
Magic squares are grids of a square shape in which the sum of the elements of each row, column, and diagonal add up to the same number. The magic square is filled in using each number from to n^2 once, where n is the number of spaces in each row of the square. In magic square puzzles, some of the blanks of a magic square puzzle are filled in for you. You must fill in the rest of the magic square so that it meets the conditions stated above. A well-written magic square puzzle only has one solution.
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Instructions
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How to Solve a Magic Square
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Part of the definition of a magic square is that each column, row, and diagonal adds up to the same number. Knowing this number before you attempt to fill in the elements of the magic square is extremely beneficial. To calculate this number for any n by n magic square, sum all the perfect squares from 1 to n^2. Because there are n rows in the square and each row adds up to the same number, the sum of each row must be (1+2^2...+n^2)/n. This formula works for every magic square.
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Now that you have determined the common sum of the rows, columns, and diagonals, it is time to start filling it in. Search the puzzle for easy squares to fill in such as 1 square left blank in a column where all the other squares are filled in. Fill in any other squares where you can deduce which number must be in them immediately just by looking at the puzzle.
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Create a chart with all the positive integers from 1 to n^2 inclusive written on it. Cross out all numbers that have already been used in the puzzle. This will be your reference as to which numbers you can use to fill in the puzzle. Whenever you fill in a blank, cross out the number that you used up. Remember that each number can only be used once.
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Solving the rest of the puzzle requires deduction and trial-and-error experimentation. For each square of the puzzle, write all the numbers that can be placed inside each square. Try to spot arrangements that may give you insight into the solution of the puzzle. If you see a number that seems like it would work inside a certain square, fill it in tentatively and attempt to solve the puzzle assuming that number works. If there is a contradiction, you can eliminate that number as a possible solution for the square in question. Repeat this process with each square of the puzzle until you have filled in every blank.
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References
- Photo Credit Sudoku image by Claude Wangen from Fotolia.com
