Magic Squares are among the most ancient puzzles known to mathematicians. A 3000-year-old Chinese folk tale includes the earliest known example of this puzzle. The basic idea of the puzzle is very simple: Use all the numbers from 1 to 9. Arrange them in a 3 x 3 square so that every row, column and diagonal adds up to the same number.
Put these numbers:
1, 2, 3, 4, 5, 6, 7, 8 9
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into this square:
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so that all the rows, columns and diagonals add up to the same number. |
This problem was used for a Puzzle of the Month in April 2001. This time we will consider some variations and extensions. You could solve the problem by trial and error, trying different numbers and adding them and switching them around. You could also use algebra and careful logic to come up with an answer. Here’s a slightly different approach: solve a simpler problem first.
 Use the numbers from 1 to 9. Place one in each box so that by adding three numbers along all the lines going through the center box, the sums are the same. Each number must be used exactly once.
Use what you learned by solving puzzle 1 to solve the magic square puzzle given above.
Can you fill a 3 x 3 magic square using the first 9 consecutive even numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18?
How about the first 9 consecutive odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17?
Both of these should be easy once you’ve solved puzzles 1 and 2.
Try to find another set of nine different numbers that can form a 3 x 3 magic square.
Here’s a harder problem. From these examples, you might get the idea that you can construct a 3 x 3 magic square using any set of different numbers. Is this true? See if you can prove it false by finding a counter example: a set of 9 different numbers that cannot form a magic square.
Try to form a 4 x 4 magic square using consecutive integers from 1 to 16.
Are there any numbers you can use to form a 2 x 2 magic square? Can you prove that a 2 x 2 magic square is impossible?
Next month we’ll take this idea further with magic stars and magic pentagons
Magic Squares and similar puzzles have been known for centuries. In addition to inspiring puzzle makers and puzzle solvers, they have also been of interest to professional mathematicians. The mathematics involves both algebra and geometry. The algebra is obvious: just try to solve magic squares using simultaneous equations, although once you get beyond 3 x 3 squares, the algebra can get complicated and difficult.
The geometry comes in when you try to compare solutions to Magic Squares to see if they are the same. For example, if a 3 x 3 magic square has a solution represented by the letters A – I, for example:
Then you can get additional solutions by rotating the entire square 90 degrees to get the square at the left below, or flipping it about its center line to get the square on the right. Both of the following squares are also correct solutions.
In fact, a mathematician would say that they are the “same” solution, because you don’t really have to rearrange the numbers, just flip or rotate the entire square.
The Drexel Math Forum has a series of pages devoted to the history and mathematics of magic squares and related shapes. The page includes several lesson plans—but even more interesting are the many links to other web sites, which reveal the mathematical complexity behind such a simple looking puzzle.
After you’ve tried this for yourself check our solutions. |
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