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At Right Angles | Vol. 3, No. 2, July 2014 31
in the classroom
From magic
squares to magic
carpets
A. Ramachandran
Magic squares are a topic of interest to mathematicians,
puzzlers and lay people alike. Apart from the mathematical
properties, mystical qualities are often attributed to these in
different cultures.
Magic squares are arrays of numbers (usually from 1 onwards)
whose rows, columns and diagonals add up to the same ‘magic’ total.
There is essentially just one 3 x 3 magic square with a magic sum of
15, but one could have obtained others by reflections and rotations.
There are a large number of different 4 x 4 magic squares (even
excluding reflections and rotations.) In several of these, apart from
the rows, columns and diagonals yielding the magic sum of 34, many
other symmetrically located quartets of numbers give the same total.
Some 4 x 4 magic squares have the property that pairs of numbers
symmetrically placed about the centre of the grid add up to 17.
Two such pairs of numbers would form an interesting pattern,
yielding the magic sum of 34. Let us for short refer to this property
as the ‘inversion symmetry’.
Math Exotica ...
Keywords: Magic squares, pattern, symmetry, inversion, quadrilaterals
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32 At Right Angles | Vol. 3, No. 2, July 2014
Source: http://en.wikipedia.org/wiki/File:Melencolia_I_
(Durero).jpg
One such 4 x 4 magic square features in a
celebrated work of art – an engraving titled
Melancholia, executed by the German artist
Albrecht Dürer in 1514. The square itself is shown
below.
Observe that this has inversion symmetry. A
straight line segment connecting the centres of a
pair of squares thus related passes through the
centre of symmetry and is bisected by it. With two
such pairs of squares, therefore, we get two line
segments that bisect each other. Hence the centres
of the four squares in question form the corners
of a parallelogram. To obtain such a parallelogram
we must choose two squares out of the eight in
one half of the 4 x 4 grid. The matching squares
(their ‘mates’) get selected automatically. Now
we have 28 ways of choosing 2 objects out of 8.
One can identify these 28 parallelograms (with
centres at the centre of the grid) and thereby
obtain 28 quartets of numbers giving the magic
total. Four of these shapes are actually squares,
while four others are non-square rectangles, two
are non-square rhombuses, sixteen are general
parallelograms, and two are straight lines (the
diagonals) which can be considered collapsed
or ‘degenerate’ parallelograms. Some of these
parallelograms are displayed below.
One such 4x4 magic square features in a celebrated work of art – an engraving titled Melancholia,
executed by the German artist Albrecht Dürer in 1514. The square itself is shown below.
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
Observe that this has inversion symmetry. A straight line segment connecting the centres of a pair
of squares thus related passes through the centre of symmetry and is bisected by it. With two
such pairs of squares, therefore, we get two line segments that bisect each other. Hence the
centres of the four squares in question form the corners of a parallelogram. To obtain such a
parallelogram we must choose two squares out of the eight in one half of the 4x4 grid. The
matching squares (their ‘mates’) get selected automatically. Now we have 28 ways of choosing 2
objects out of 8. One can identify these 28 parallelograms (with centres at the centre of the grid)
and thereby obtain 28 quartets of numbers giving the magic total. Four of these shapes are
actually squares, while four others are non-square rectangles, two are non-square rhombuses,
sixteen are general parallelograms, and two are straight lines (the diagonals) which can be
considered collapsed or ‘degenerate’ parallelograms. Some of these parallelograms are displayed
below.
10 11
6 7
16 13
4 1
3
8
9
14
16
11
6
1
The equality of row and column sums is not a consequence of the inversion symmetry. They are
independently contrived by a judicious distribution of the numbers 1 to 8 in the grid. (The other
numbers then get assigned automatically.)
Each row and each column shares a symmetry axis with the entire grid. Eight other quartets
giving the magic sum and sharing a symmetry axis with the entire grid are as follows: the four
squares in each quadrant of the main grid and the corners of four 3x3 squares.
One such 4x4 magic square features in a celebrated work of art – an engraving titled Melancholia,
executed by the German artist Albrecht Dürer in 1514. The square itself is shown below.
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
Observe that this has inversion symmetry. A straight line segment connecting the centres of a pair
of squares thus related passes through the centre of symmetry and is bisected by it. With two
such pairs of squares, therefore, we get two line segments that bisect each other. Hence the
centres of the four squares in question form the corners of a parallelogram. To obtain such a
parallelogram we must choose two squares out of the eight in one half of the 4x4 grid. The
matching squares (their ‘mates’) get selected automatically. Now we have 28 ways of choosing 2
objects out of 8. One can identify these 28 parallelograms (with centres at the centre of the grid)
and thereby obtain 28 quartets of numbers giving the magic total. Four of these shapes are
actually squares, while four others are non-square rectangles, two are non-square rhombuses,
sixteen are general parallelograms, and two are straight lines (the diagonals) which can be
considered collapsed or ‘degenerate’ parallelograms. Some of these parallelograms are displayed
below.
10 11
6 7
16 13
4 1
3
8
9
14
16
11
6
1
The equality of row and column sums is not a consequence of the inversion symmetry. They are
independently contrived by a judicious distribution of the numbers 1 to 8 in the grid. (The other
numbers then get assigned automatically.)
Each row and each column shares a symmetry axis with the entire grid. Eight other quartets
giving the magic sum and sharing a symmetry axis with the entire grid are as follows: the four
squares in each quadrant of the main grid and the corners of four 3x3 squares.
One such 4x4 magic square features in a celebrated work of art – an engraving titled Melancholia,
executed by the German artist Albrecht Dürer in 1514. The square itself is shown below.
16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1
Observe that this has inversion symmetry. A straight line segment connecting the centres of a pair
of squares thus related passes through the centre of symmetry and is bisected by it. With two
such pairs of squares, therefore, we get two line segments that bisect each other. Hence the
centres of the four squares in question form the corners of a parallelogram. To obtain such a
parallelogram we must choose two squares out of the eight in one half of the 4x4 grid. The
matching squares (their ‘mates’) get selected automatically. Now we have 28 ways of choosing 2
objects out of 8. One can identify these 28 parallelograms (with centres at the centre of the grid)
and thereby obtain 28 quartets of numbers giving the magic total. Four of these shapes are
actually squares, while four others are non-square rectangles, two are non-square rhombuses,
sixteen are general parallelograms, and two are straight lines (the diagonals) which can be
considered collapsed or ‘degenerate’ parallelograms. Some of these parallelograms are displayed
below.
10 11
6 7
16 13
4 1
3
8
9
14
16
11
6
1
The equality of row and column sums is not a consequence of the inversion symmetry. They are
independently contrived by a judicious distribution of the numbers 1 to 8 in the grid. (The other
numbers then get assigned automatically.)
Each row and each column shares a symmetry axis with the entire grid. Eight other quartets
giving the magic sum and sharing a symmetry axis with the entire grid are as follows: the four
squares in each quadrant of the main grid and the corners of four 3x3 squares.
Page 3 of 3
At Right Angles | Vol. 3, No. 2, July 2014 33
The equality of row and column sums is not a
consequence of the inversion symmetry. They
are independently contrived by a judicious
distribution of the numbers 1 to 8 in the grid. (The
other numbers then get assigned automatically.)
Each row and each column shares a symmetry
axis with the entire grid. Eight other quartets
giving the magic sum and sharing a symmetry
axis with the entire grid are as follows: the four
squares in each quadrant of the main grid and the
corners of four 3 x 3 squares.
There are sixteen other quartets giving the magic
sum, and with at least one element of symmetry,
but not placed symmetrically in the main grid.
These are eight 2 x 3 rectangles, four ‘kites’ (2
erect and 2 inverted) and four arrowheads (2
erect and 2 inverted). The latter two patterns
appear only in the vertical sense and have no
horizontal counterparts.
There are 86 ways of obtaining a sum of 34 by
choosing four numbers from 1-16. (It would be
an interesting but challenging exercise for the
student to verify this.) Durer’s magic square
exhibits 60 of these in symmetrical patterns. (The
student is invited to verify this as well.)
An Indian magic square
As a counterpoint to the magic square discussed
above we look at a magic square of Indian origin.
It does not have the inversion property and so
does not exhibit many properties that follow
from it. However, apart from the row, column and
diagonal property it has other interesting features.
Quartets of numbers forming the corners of eight
isosceles trapeziums add to the magic sum. An
example is given below.
It has the ‘pandiagonal’ property, that is, quartets
formed from numbers on the broken diagonals
give the magic sum. An example follows.
Every 2 x 2 square gives the magic sum, as does
the set of corners of each 2 x 4 rectangle.
The magic square can be extended by repetition in
both East-West and North-South directions to give
a ‘Magic carpet’ – an open 2-D array of numbers,
where any four neighbouring numbers in a line
(vertical, horizontal or diagonal) or forming a 2 x 2
square yield the magic sum. In addition, numbers
at the corners of any 3 x 3 square and any 4 x 4
square yield the magic sum.
Further investigations in this area will surely
prove to be a ‘magic carpet ride’ for a young
mathematician or puzzle enthusiast.
A RAMACHANDRAN has had a long standing interest in the teaching of mathematics and science. He studied
physical science and mathematics at the undergraduate level, and shifted to life science at the postgraduate level.
He taught science, mathematics and geography to middle school students at Rishi Valley School for over two
decades, and now stays in Chennai. His other interests include the English language and Indian music. He may be
contacted at archandran.53@gmail.com.
There are sixteen other quartets giving the magic sum, and with at least one element of symmetry,
but not placed symmetrically in the main grid. These are eight 2x3 rectangles, four ‘kites’ (2 erect
and 2 inverted) and four arrowheads (2 erect and 2 inverted). The latter two patterns appear only
in the vertical sense and have no horizontal counterparts.
There are 86 ways of obtaining a sum of 34 by choosing four numbers from 1-16. (It would be an
interesting but challenging exercise for the student to verify this.) Durer’s magic square exhibits
60 of these in symmetrical patterns. (The student is invited to verify this as well.)
AN INDIAN MAGIC SQUARE
As a counterpoint to the magic square discussed above we look at a magic square of Indian origin.
7 12 1 14
2 13 8 11
16 3 10 5
9 6 15 4
It does not have the inversion property and so does not exhibit many properties that follow from
it. However, apart from the row, column and diagonal property it has other interesting features.
Quartets of numbers forming the corners of eight isosceles trapeziums add to the magic sum. An
example is given below.
12 1
16 5
It has the ‘pandiagonal’ property, that is, quartets formed from numbers on the broken diagonals
give the magic sum. An example follows.
1
13
16
4
Every 2x2 square gives the magic sum, as does the set of corners of each 2x4 rectangle.
There are sixteen other quartets giving the magic sum, and with at least one element of symmetry,
but not placed symmetrically in the main grid. These are eight 2x3 rectangles, four ‘kites’ (2 erect
and 2 inverted) and four arrowheads (2 erect and 2 inverted). The latter two patterns appear only
in the vertical sense and have no horizontal counterparts.
There are 86 ways of obtaining a sum of 34 by choosing four numbers from 1-16. (It would be an
interesting but challenging exercise for the student to verify this.) Durer’s magic square exhibits
60 of these in symmetrical patterns. (The student is invited to verify this as well.)
AN INDIAN MAGIC SQUARE
As a counterpoint to the magic square discussed above we look at a magic square of Indian origin.
7 12 1 14
2 13 8 11
16 3 10 5
9 6 15 4
It does not have the inversion property and so does not exhibit many properties that follow from
it. However, apart from the row, column and diagonal property it has other interesting features.
Quartets of numbers forming the corners of eight isosceles trapeziums add to the magic sum. An
example is given below.
12 1
16 5
It has the ‘pandiagonal’ property, that is, quartets formed from numbers on the broken diagonals
give the magic sum. An example follows.
1
13
16
4
Every 2x2 square gives the magic sum, as does the set of corners of each 2x4 rectangle.
There are sixteen other quartets giving the magic sum, and with at least one element of symmetry,
but not placed symmetrically in the main grid. These are eight 2x3 rectangles, four ‘kites’ (2 erect
and 2 inverted) and four arrowheads (2 erect and 2 inverted). The latter two patterns appear only
in the vertical sense and have no horizontal counterparts.
There are 86 ways of obtaining a sum of 34 by choosing four numbers from 1-16. (It would be an
interesting but challenging exercise for the student to verify this.) Durer’s magic square exhibits
60 of these in symmetrical patterns. (The student is invited to verify this as well.)
AN INDIAN MAGIC SQUARE
As a counterpoint to the magic square discussed above we look at a magic square of Indian origin.
7 12 1 14
2 13 8 11
16 3 10 5
9 6 15 4
It does not have the inversion property and so does not exhibit many properties that follow from
it. However, apart from the row, column and diagonal property it has other interesting features.
Quartets of numbers forming the corners of eight isosceles trapeziums add to the magic sum. An
example is given below.
12 1
16 5
It has the ‘pandiagonal’ property, that is, quartets formed from numbers on the broken diagonals
give the magic sum. An example follows.
1
13
16
4
Every 2x2 square gives the magic sum, as does the set of corners of each 2x4 rectangle.