|
What is a Magic 3x3 Square?
..............8....1....6.........................
..............3....5....7.........................
..............4....9....2......................... |
You can assemble the numbers 1 to 9 in a square, so that the sum of
the rows, the columns, and the diagonals is 15.
If you take the numbers 1 to 9, you have the standard square. |
.............8+c....1+c....6+c..............
.............3+c....5+c....7+c..............
.............4+c....9+c....2+c.............. |
A magic square remains magic, if you change each numbers by a constant
c. You add c on the left. You can also subtract, multiply or divide. |
You define 4x4-, 5x5-... squares correspondingly.
The Magic 3x3 Square top
You have 1+2+3+4+5+6+7+8+9=45. In a magic square you have to add 3
numbers again and again. Therefore the average sum of three numbers is
45:3=15. The number 15 is called the magic number of the 3x3 square.
You can also achieve 15, if you add the middle number 5 three times.
You can reduce 15 in a sum of three summands eight times:
15=1+5+9
15=1+6+8
|
15=2+4+9
15=2+5+8
|
15=2+6+7
15=3+4+8
|
15=3+5+7
15=4+5+6
|
The odd numbers 1,3,7, and 9 occur twice in the reductions, the even numbers
2,4,6,8 three times and the number 5 four times.
Therefore you have to place number 5 in the middle of the magic 3x3
square. The remaining odd numbers have to be in the middles of a side and
the even numbers at the corners.
Under these circumstances there are eight possibilities building a
square:
All the eight squares change into each other, if you reflect them at the
axes of symmetry. You count symmetric squares only once. Therefore there
is only one magic 3x3 square.
The Magic 4x4 Square top
The magic number is (1+2+...+15+16):4
= 34.
The computer found 86 reductions of 34 to a sum of four summands with
the numbers 1 to 16.
34=01+02+15+16
34=01+03+14+16
34=01+04+13+16
34=01+04+14+15
34=01+05+12+16
34=01+05+13+15
34=01+06+11+16
34=01+06+12+15
34=01+06+13+14
34=01+07+10+16
34=01+07+11+15
34=01+07+12+14
34=01+08+09+16
34=01+08+10+15
34=01+08+11+14
34=01+08+12+13
34=01+09+10+14
34=01+09+11+13 |
34=01+10+11+12
34=02+03+13+16
34=02+03+14+15
34=02+04+12+16
34=02+04+13+15
34=02+05+11+16
34=02+05+12+15
34=02+05+13+14
34=02+06+10+16
34=02+06+11+15
34=02+06+12+14
34=02+07+09+16
34=02+07+10+15
34=02+07+11+14
34=02+07+12+13
34=02+08+09+15
34=02+08+10+14
34=02+08+11+13 |
34=02+09+10+13
34=02+09+11+12
34=03+04+11+16
34=03+04+12+15
34=03+04+13+14
34=03+05+10+16
34=03+05+11+15
34=03+05+12+14
34=03+06+09+16
34=03+06+10+15
34=03+06+11+14
34=03+06+12+13
34=03+07+08+16
34=03+07+09+15
34=03+07+10+14
34=03+07+11+13
34=03+08+09+14
34=03+08+10+13 |
34=03+08+11+12
34=03+09+10+12
34=04+05+09+16
34=04+05+10+15
34=04+05+11+14
34=04+05+12+13
34=04+06+08+16
34=04+06+09+15
34=04+06+10+14
34=04+06+11+13
34=04+07+08+15
34=04+07+09+14
34=04+07+10+13
34=04+07+11+12
34=04+08+09+13
34=04+08+10+12
34=04+09+10+11
34=05+06+07+16 |
34=05+06+08+15
34=05+06+09+14
34=05+06+10+13
34=05+06+11+12
34=05+07+08+14
34=05+07+09+13
34=05+07+10+12
34=05+08+09+12
34=05+08+10+11
34=06+07+08+13
34=06+07+09+12
34=06+07+10+11
34=06+08+09+11
34=07+08+09+10
.
.
.
. |
The summands 1 to 16 are distributed regularly in the reductions:
Summand:
Number: |
01
19 |
02
20 |
03
21 |
04
22 |
05
22 |
06
23 |
07
23 |
08
22 |
09
22 |
10
23 |
11
23 |
12
22 |
13
22 |
14
21 |
15
20 |
16
19 |
Unlike the 3x3 square there is not just one conclusion for the distribution
of the numbers 1 to 16 in a 4x4 square.
Fact: There are 880 magic squares, counting the symmetric ones only
once.
This is one of 880 possible squares:
............12....06....15.....01............
............13....03....10.....08............
............02....16....05.....11............
............07....09....04.....14............ |
This square is special. The number 34 is not only the sum of the numbers
in the rows, the columns, and both diagonals, but also in every 2x2 square. |
The Magic 5x5 Square top
The magic number is (1+2+...+24+25) : 5 = 65.
Reductions of the magic number 65.
65 = 01+02+13+24+25
65 = 01+02+14+23+25
65 = ...
65 = ... |
65 = ...
65 = ...
65 = 10+12+13+14+16
65 = 11+12+13+14+15 |
The computer found 1394 reductions of the number 65. |
The summands and their number in the sums
You notice, that the middle number 13 = 65:5 appears most frequently. The
numbers of the summands to smaller and bigger summands drop symmetrically
to both sides.
Fact: There are 275 305 224 magic 5x5 squares. (Scientific American
1/1976)
Making of a magic 5x5-square:
You go through the numbers 1 to 25. There are two rules for constructing
a magic square "top right" and "if the place is occupied, go one down".
..................... .................... |
Number 1 is placed in the centre of the
first row.
Number 2 follows top right. But then you
leave the 5x5 square.
Therefore you must imagine the square is a cylinder. The cylinder has the
vertical square sides of the square as circumference. The horizontal sides
touch each other and close the curved surface of the cylinder. So there
is a field top right for number 2. If you unroll the cylinder, number 2
has gone to the last row one place to the right. |
Number 3 follows top right.
... ... |
Number 4 would lie outside the 5x5 square.
So again you imagine the square would become a cylinder, this time with
vertical axis. You can find a place for number 4. If you unroll the cylinder,
you find the number in the third row on the far left. |
Number 5 lies top right.
You use the second rule for number 6.
If the field top right is occupied, you put the number one row down in
the same column.
You go further on with 7, 8,
and so on. - You use the same rule for number 16
as for number 6.
You can transfer this way of formation to
all magic squares with odd numbers of the sides ;-).
There are also rules for magic squares with an even side length. They
are more complicated however.
The Magic nxn Square top
The existence of magic squares nxn is proved for all numbers n>2. But
there is no general rule.
The magic number is (1 + 2 + 3 + ... + n²) : n =0.5 * (n²+1)
* n.
The magic numbers of the standard squares:
Magic squares: 3x3 4x4
5x5 6x6 7x7
8x8 9x9 10x10
Magic numbers: 15 34
65 111 175
260 369 505
Curiosities top
 |
Once again: A square is magic, if the numbers have the same
sum in the rows, the columns and the diagonals. |
... ...
|
A square is semi-magic, if the numbers have the same sum only
in the rows and the columns. |
... ...
|
The Latin square, which is formed by 1,2, and 3 and has different numbers
in every row and column, is an example of a semi-magic square. |
 ... |
A square is panmagic, if it is magic and if the numbers have
the same sum in the corners, in the centre and on the edges. |
|
The Dürer square is panmagic. |
 ... |
A square is pandiagonal, if it is magic and if not only the
numbers of the main diagonals, but also the broken diagonals have the same
sum. |
|
The magic square is pandiagonal (and panmagic). |
... ... |
A square is complementary, if every number n is replaced by
17-n and it stays magic. |
Formula: Replace n in any n x n-square by (n²+1)-n.
The example is self-complemetary, because the new square is symmetric
to the old one. See red axis.
 ... |
A square is associative, if it is magic and if pairs of numbers
lying symmetrically to the centre have the same sum. The sum is 26=5²+1
=n²+1. |
|
The 3x3 magic square Lo Shu is associative. |
You can fix the kind of the numbers.
|
A magic square with prime numbers |
|
A "semi"-magic square of square numbers
All sums going through the centre have the same value.
Source: http://www.mathpages.com/home/kmath417.htm |
If a magic square is panmagic, pandiagonal and
self-complementary (and has more patterns), it is ultra-super-magic
(Mutsumi Suzuki).
The sequence of these special magic squares
is not finished. There are magic squares missing e.g. formed by smaller
magic squares or bordered squares (Book 2).
There is an explanation for all the different magic squares: They can
be found by computers. A new property means a new query inside the program.
Simple Variants
top
Magic Squares on the
Internet top
English
Craig Knecht (Magic
Square Models)
| Water
Retention Patterns |
... ... |
Imagine, the squares are the top of square prisms with the height given
by the numbers. If you pour water in this solid, it stays in the centre
upto the height 17. Then it flows off. The amount of water is (17-3)+(17-7)+(17-13)+(17-1)+(17-4)+(17-5)=69.
There are nice problems: Biggest amount of water? Seperate ponds? Island? |
Eric W. Weisstein (MathWorld)
Magic Square,
Panmagic
Square,
Associative
Magic Square, Lo
Shu
Harvey D. Heinz
Magic Squares, Magic Stars
& Other Patterns
Ivars Peterson's MathTrek
More than
Magic Squares
Mark S. Farrar
Magic Squares
MathPages
Solving Magic
Squares
Suzanne Alejandre (Math Forum)
Magic squares,
Mutsumi Suzuki's Pages
included
Wikipedia
Magic square,
Most-perfect
magic square
German
Feng-Shui-Homepage
Das magische Quadrat Lo-Shu
Hans-Peter Gramatke
Magische Quadrate
Jan Haase
Das
Hexeneinmaleins aus Goethes "Faust" (Lösung)
Jan Theofel und Martin Trautmann
Magische Quadrate
und Würfel
Maria Koth
Magische
Quadrate (.pdf.-Datei)
Paul Heimbach
Magische Quadrate
recordholders.org
Das
größte Magische Quadrat der Welt
Udo Hebisch (Mathematisches Café)
Magische
Quadrate
Wikipedia
Magisches
Quadrat, Vollkommen
perfektes magisches Quadrat
References top
(1) Bild der Wissenschaften, Heft 8/1966, Heft 6/1968, Heft 10/1976
(2) Pieter van Delft /Jack Botermans: Denkspiele der Welt, München
1980 (1998 neu aufgelegt)
(3) Maximilian Miller, Gelöste und ungelöste mathematische
Probleme, Leipzig 1982
Feedback: Email address on my main page
This
page is also available in German.
URL of
my Homepage:
http://www.mathematische-basteleien.de/
©
2000 Jürgen Köller
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