2. Source of the image here
What is a magic square?What is a magic square?
3. all numbers from 1 to 9 areall numbers from 1 to 9 are
used without repeating anyused without repeating any
of them.of them.
the sum of these numbersthe sum of these numbers
regardless of the vertical,regardless of the vertical,
horizontal or diagonalhorizontal or diagonal
direction is always 15.direction is always 15.
The number obtained by adding the numbers in theThe number obtained by adding the numbers in the
square horizontally, vertically or diagonally is called thesquare horizontally, vertically or diagonally is called the
constant of the magic square. In our case, the constantconstant of the magic square. In our case, the constant
of the magic square is 15of the magic square is 15
Another exampleAnother example
4. Some magic squares in historySome magic squares in history
Old Chinese magic square
Magic squares as amulets
Famous square in Durer’s painting
5. How to make a magic squareHow to make a magic square
1+2+3+4+5+6+7+8+9=45. In a1+2+3+4+5+6+7+8+9=45. In a
magic square you have tomagic square you have to
repeatedly add 3 numbers.repeatedly add 3 numbers.
Therefore the average sum of threeTherefore the average sum of three
numbers is 45:3=15. Thenumbers is 45:3=15. The
numbernumber 1515 is called the magicis called the magic
number (or constant) of the square.number (or constant) of the square.
You can also achieve 15, if you addYou can also achieve 15, if you add
the middle number 5 three times.the middle number 5 three times.
You can reduce 15 in a sum of threeYou can reduce 15 in a sum of three
numbers eight times. Write themnumbers eight times. Write them
down!down!
6. How to make a magic squareHow to make a magic square
The odd numbers 1,3,7, and 9 occur twice inThe odd numbers 1,3,7, and 9 occur twice in
the reductions, the even numbers 2,4,6,8the reductions, the even numbers 2,4,6,8
three times and the number 5 four times.three times and the number 5 four times.
Therefore you have to place number 5 in theTherefore you have to place number 5 in the
middle of the magic 3x3 square. Themiddle of the magic 3x3 square. The
remaining odd numbers have to be in theremaining odd numbers have to be in the
middles of a side and the even numbers atmiddles of a side and the even numbers at
the corners.the corners.
Under these circumstances there are eightUnder these circumstances there are eight
possibilities building a square. Find them!possibilities building a square. Find them!
All the eight squares change into each other,All the eight squares change into each other,
if you reflect them at the axes of symmetry.if you reflect them at the axes of symmetry.
You count symmetric squares only once.You count symmetric squares only once.
Therefore there is only one magic 3x3Therefore there is only one magic 3x3
square.square.
7. TasksTasks
1.1. Fill in the missing numbers in the 4x4 squaresFill in the missing numbers in the 4x4 squares
on your worksheet.on your worksheet.
2.2. Create all the eight 3x3 magic squares. FindCreate all the eight 3x3 magic squares. Find
examples of rotations and reflections.examples of rotations and reflections.
3.3. Find the formula for the constant of the 3x3,Find the formula for the constant of the 3x3,
4x4 and nxn magic square with consecutive4x4 and nxn magic square with consecutive
numbers.numbers.
4.4. Create one 4x4 magic square. For every 2x2Create one 4x4 magic square. For every 2x2
square inside yours, calculate its constant.square inside yours, calculate its constant.
9. TasksTasks
6.6. Create a magic square with 10 in the middleCreate a magic square with 10 in the middle
position. Explain how you did it.position. Explain how you did it.
7.7. Create a magic square with the first 9 oddCreate a magic square with the first 9 odd
numbers.numbers.
8.8. The same with the first 9 even numbers. CanThe same with the first 9 even numbers. Can
you use the previous one as help?you use the previous one as help?
9.9. Fill in the fractions magic squares on yourFill in the fractions magic squares on your
worksheet.worksheet.
10.10.Fill in the decimal magic squares on yourFill in the decimal magic squares on your
worksheet.worksheet.
