Magic SquaresMagic SquaresMathematical/Logical Intelligence Mathematical/Logical Intelligence

activityactivity

Source of the image here

What is a magic square? What is a magic square?

all numbers from 1 to 9 are all numbers from 1 to 9 are used without repeating any used without repeating any of them. of them.

the sum of these numbers the sum of these numbers regardless of the vertical, regardless of the vertical, horizontal or diagonal horizontal or diagonal direction is always 15.direction is always 15.

The number obtained by adding the numbers in the The number obtained by adding the numbers in the square horizontally, vertically or diagonally is called the square horizontally, vertically or diagonally is called the constant of the magic square. In our case, the constant constant of the magic square. In our case, the constant of the magic square is 15of the magic square is 15

Another exampleAnother example

Some magic squares in historySome magic squares in history

Old Chinese magic square

Magic squares as amuletsFamous square in Durer’s painting

How to make a magic squareHow to make a magic square

1+2+3+4+5+6+7+8+9=45. In a 1+2+3+4+5+6+7+8+9=45. In a magic square you have to magic square you have to repeatedly add 3 numbers. repeatedly add 3 numbers. Therefore the average sum of three Therefore the average sum of three numbers is 45:3=15. The numbers is 45:3=15. The number number 1515 is called the magic is called the magic number (or constant) of the square. number (or constant) of the square. You can also achieve 15, if you add You 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 three You can reduce 15 in a sum of three numbers eight times. Write them numbers eight times. Write them down!down!

How to make a magic squareHow to make a magic square The odd numbers 1,3,7, and 9 occur twice in The odd numbers 1,3,7, and 9 occur twice in

the reductions, the even numbers 2,4,6,8 the 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 the Therefore you have to place number 5 in the middle of the magic 3x3 square. The middle of the magic 3x3 square. The remaining odd numbers have to be in the remaining odd numbers have to be in the middles of a side and the even numbers at middles of a side and the even numbers at the corners.the corners.

Under these circumstances there are eight Under 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 3x3 Therefore there is only one magic 3x3 square. square.

TasksTasks

1.1. Fill in the missing numbers in the 4x4 squares Fill in the missing numbers in the 4x4 squares on your worksheet. on your worksheet.

2.2. Create all the eight 3x3 magic squares. Find Create 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 consecutive 4x4 and nxn magic square with consecutive numbers.numbers.

4.4. Create one 4x4 magic square. For every 2x2 Create one 4x4 magic square. For every 2x2 square inside yours, calculate its constant. square inside yours, calculate its constant.

TasksTasks

A magic square resulting in another one (source)

TasksTasks6.6. Create a magic square with 10 in the middle Create 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 odd Create a magic square with the first 9 odd

numbers.numbers.8.8. The same with the first 9 even numbers. Can The 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 your Fill in the fractions magic squares on your

worksheet. worksheet. 10.10. Fill in the decimal magic squares on your Fill in the decimal magic squares on your

worksheet. worksheet.

Advanced task Advanced task

Source of the image here

Well done! Well done!