1. What is a Magic Square?
Magic square in mathematics is one of the most interesting puzzles with a very special arrangement of numbers in the square grids. The very special rule of the magic square which makes it interesting is the sum of the numbers along each row, column and diagonal is the same. Look at the 3 x 3 magic square shown below:
Sum of the numbers along the rows:
Row 1: 2 + 9 + 4 = 15,
Row 2: 7 + 5 + 3 = 15,
Row 3: 6 + 1 = 8 = 15,
Sum of the numbers along the columns:
Column 1: 2 + 7 + 6 = 15
Column 2: 9 + 5 + 1 = 15
Column 3: 4 + 3 + 8 = 15
Sum of the numbers along the diagonals:
Diagonal 1: 2 + 5 + 8 = 15
Diagonal 2: 4 + 5 + 6 = 15
As you can see, the sum of the numbers along each row, column and diagonal is the same. This is the property of the magic squares.
1.1 How to Solve Magic Squares?
In the 11 Plus exams, you might be given an incomplete magic square and asked to complete it by filling in the missing positions. There are nine positions to be filled with numbers in 3 x 3 magic squares, but for 4 x 4 magic squares, the number of positions increases to 16 and the complexity to arrange the numbers increases. In such cases, figuring out the common sum for the magic square can ease the working.
Use the following steps to solve questions on Magic Squares:
- Look for a row, column or diagonal with one missing number.
- Then, try to find that missing number by subtracting the sum of the remaining numbers from the magic number given.
- Repeat the first two steps until you find all the missing numbers.
If the common sum for the magic square is not given:
- Fill out the missing positions with the algebraic variables.
- Add up the numbers and variables across the rows/columns/diagonals.
- Equate the expressions formed and solve for the variables.
Let us now apply the above steps and solve the magic square problems using the examples.
1.2 Solved Examples on Magic Squares
Example 1: The magic square is shown below. All the rows, columns and diagonals add up to 21.
Which number replaces A?
Solution:
First, find the missing number on the diagonal containing 7 and 8.
Missing number = 21 - (8 + 7) = 6
For the third row: B = 21 - (6 + 4) = 11
For the second column: A = 21 - (7 + B)
A = 21 - (7 + 11) = 3
Example 2: The diagram below shows part of a magic square.
What is the value of m?
Solution:
Here, the common sum is not given, so filling up the missing positions with algebraic variables.
Sum along row 1: a + 1 + 6 = a + 7
Sum along column 1: a + b + 2
As the sum is same along the rows and columns, equating the above expressions:
a + 7 = a + b + 2
b = a + 7 - a - 2
b = 7 - 2 = 5
Sum along row 2: b + c + m = 5 + c + m
Sum along diagonal from the right: 6 + c + 2 = 8 + c
Equating the sum:
5 + c + m = 8 + c
m = 8 + c - c - 5
m = 8 - 5 = 3
Hence, the value of m is 3.
