Adding Successive Numbers

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Adding Successive Numbers
1 2 3 4 5 6
7 8 9 10
Sum (1 ? 10) 5 x 11 55
1 2 3 18 19
20
Sum (1 ? 20) 10 x 21 210
1 2 3 98 99
100
Sum (1 ? 100) 50 x 101 5050
Generalising
1 2 3
n
n-1
n-2
3
5
1
15
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4 Reflections
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In the Middle Ages magic squares were believed to
give protection against the plague! In the 16th
Century, the Italian mathematician, Cardan, made
an extensive study of the properties of magic
squares and in the following century they were
extensively studied by several leading Japanese
mathematicians. During this century they have
been used as amulets in India, as well as been
found in oriental fortune bowls and medicine
cups. Even today they are widespread in Tibet,
(appearing in the Wheel of Life) and in other
countries such as Malaysia, that have close
connections with China and India.
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2
3
8
5
12
9
15
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16
16
2
13
2
13
3
3
16
2
13
3
5
10
5
10
11
8
11
8
10
5
8
11
12
12
9
7
9
7
6
6
7
12
9
6
4
1
4
1
15
14
15
14
4
14
1
15
16
2
13
3
16
2
13
3
16
2
13
3
10
5
8
11
10
5
8
11
5
10
11
8
7
12
9
6
7
12
9
6
12
9
7
6
4
14
1
15
4
14
1
15
4
1
15
14
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Constructing a 4 x 4 Magic Square
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A 4 x4 straight off
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3. Fill the holes
1. Build the pyramid
A 3 x 3 Construction
2. Fill the diagonals
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1. Build the pyramid
A 5 x 5 Construction
3. Fill the holes
2. Fill the diagonals
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Mathematicians have recently programmed a
computer to calculate the number of 5 x 5 magic
squares. There are exactly 275 305 224 distinct
solutions!
Construct a 7 x 7 magic Square!
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1. Build the pyramid
A 7 x 7 Construction
3. Fill the holes
2. Fill the diagonals
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Whats the magic number?
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Franklins 8 x 8 Magic Square
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Check the sum of the diagonals.
As in Eulers chessboard solution, the square is
not completely magic
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By interchanging rows, columns, and corner
groups, can you find some other distinct magic
squares?
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