Magic square

Magic Squares

A magic square is not really magic at all! It’s just a grid containing numbers that all add up to the same total in every direction—horizontally, vertically and diagonally. A common question in 11+ Maths exams asks you to fill out a 3 x 3 magic square, and this is how you do it…

The Magic Constant

The first thing you need to know is the ‘magic constant’, which is the total of the numbers in each row, column and diagonal. Sometimes, the exam paper will tell you what it is, but if you have to work it out yourself, the answer is 15 for a 3 x 3 grid. If it’s a bigger grid, you’ll have to use a complicated formula—sorry!—which is n(n² + 1) ÷ 2, where n is the number of rows or columns. For a 4 x 4 grid, this would be 4(4² + 1) ÷ 2 = 4 x 17 ÷ 2 = 34.

Normal and Non-normal Magic Squares

A normal magic square needs to be filled out with consecutive numbers starting from 1, so a 3 x 3 magic square would contain the numbers 1-9. However, in a non-normal magic square, the lowest number may be higher than 1. That means the magic constant of a 4 x 4 square would be higher than 34.

How to Fill out Magic Squares

Here’s a step-by-step guide to filling out a 3 x 3 magic square with the numbers 1-9:

  • Work out the magic constant (if necessary).
  • Put 1 in the middle box on the top row.
  • Fill in the remaining numbers consecutively by moving up one square and right one square each time, starting with the box containing the 1.

If you run out of room either vertically or horizontally, imagine that the grid ‘wraps around’ to the other edge. In other words, the next row above the top one is actually the bottom row, and the next column after the third one is actually the first column.

If you end up on a square that already has a number in it, put the next number in the box below the previous one, eg 4 goes below 3 and 7 goes below 6.

Another Way of Doing it

If all that sounds too complicated, it helps to understand the way the square works. If you look at the magic square at the top of this article, you might see the following patterns:

  • The middle number is 5, which is the average of the total of the three numbers in each row, column and diagonal
  • The bottom-left and top-right numbers are formed by subtracting 1 from and adding 1 to the middle number.
  • The left and right numbers in the middle row are formed by subtracting 2 from and adding 2 to the middle number.
  • The top-left and bottom-right numbers are formed by adding 3 to and subtracting 3 from the middle number.

If you can remember those patterns and the fact that 1 always goes in the middle square of the top row, it should make life easier!

 

 

 

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