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# Magic squares

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### Magic squares

1. 1. Magic SquaresMagic Squares Mathematical/Logical IntelligenceMathematical/Logical Intelligence activityactivity
2. 2. Source of the image here What is a magic square?What is a magic square?
3. 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. 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. 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. 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. 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.