Contents
The paper in which F. A. P. Barnard published the first perfect (what
we now call Nasik) magic cubes actually consisted mostly of material on
magic squares [1].
J. R. Hendricks (Canada 19292007) is considered the originator of the
Inlaid Magic Square, but here are two that appeared in Barnard’s paper.
Barnard uses the terms bordered and inlaid and
clearly defines the difference between the two types of magic
squares. If you remove the green cells of this order10 simple magic square
and close up the spaces, you end up with an order8 magic square. In
both cases, the central order4 square is pandiagonal magic. Remove the rose colored cells from this order14 magic square and
close up the array to obtain an order12 magic square. If the 5 green
4x4 squares are then removed, you obtain an order8 magic square.
Now remove all cells except those within the four 2x2 blue borders,
and close up this array to obtain an order4 pandiagonal magic
square. Finally, combine all the original 2x2 bordered arrays from
the original order14 square to obtain an order6 magic square.
The squares resulting from the changes in these two squares may be
seen here. Some of John Hendricks inlaid magic squares may be seen here. 

[1] F. A.
P. Barnard, Theory of Magic Squares and Cubes, Nat. Academy of
Sciences, Vol. IV, Sixth Memoir, 1888, pp 207270
Collison
Patchwork Squares
David M. Collison (U.S.A.
19331991) investigated generalized parts, from which he
constructed what he called patchwork magic squares.
[1]
Here I show an order14 magic square. He
constructed it from four 4 x 4 elbows, four 6 x 4 tees, one 6 x 6
cross, and four 4 x 4 squares. His square used associated order4
magic squares which I transposed in this figure, to pandiagonal
magic, by moving two bottom rows to the top, and then two left
columns to the right.
All figures sum to a
constant in proportion to the number of cells in a line. S _{14}
= 1379 so S_{2} = 197, S_{4} = 394, and S_{6}
= 591.
Because all figures are
magic, they can be rearranged, using rotations and reflections to
produce a large variety of different order14 magic squares.
[1] John R. Hendricks,
The Magic Square Course, selfpublished, 1991, 554 pages (pp
309318)
Another patchwork square
by Collison is this order7 overlapping square shown in Decimal and
Radix7 form.
The Decimal normal, simple
magic square with S = 175 contains an order3 semimagic square with
S = 75, two 2 x 4 magic rectangles with S = 50 and 100, and an
order5 pandiagonal magic square with S = 125.
The Radix7 square is the
equivalent of the above, but numbered from 00 to 66 (instead of 01
to 100). S = 330. The 3 x 3 semimagic square contains all 2digit
combinations of 0, 3, and 6. S = 132. The two 2 x 4 rectangles have
S = 66 and 165.
The pandiagonal 5 x 5 sums to 231. 

Gnomon Magic Squares
Gnomon:
the remainder of a parallelogram after the
removal of a similar parallelogram containing one of its
corners.MerriamWebster Online
A 3 x 3 array may be dissected into a 2 x 2 array and a 5 cell gnomon
in 4 ways.
When the sum of the cells in the four 2 x 2 arrays sum to the same value,
the large array is said to be a gnomonmagic square.
[1]
A. is such a square. Each of the
corner 2x2 regions sum to 20. Refer to these regions as C.
B. This gnomon square consists of the
9 smallest prime numbers. C = 33. It is possible to construct an infinite
variety of other prime magic gnomon squares
with 2 in the center cell.
The smallest prime number C of any gnomon square with 9 distinct primes
has C = 43. Can you construct it?
C.D.E.F. Here are four
gnomonantimagic squares. The four C's (of each) = 19, 21, 23, and 25.
G. The Lhoshu has a special
significance in ancient Arab culture when it is considered in this psuedognomon
manner. [2]
The Muslim alchemist, Jabir ibn Hayyan, considered the entire material
world based on 17.
He noted too, the series 1:3:5:8 formed the foundation for all other
numbers.
28 is the typical moon number. It is also the number of letters in the
Arabic (abjad) alphabet.
The 2x2 shown in green = 17, the gnomon = 28.
H. Any 3x3 magic square may be
converted to a gnomon square by rotating the perimeter numbers 1 position.
C = 20.
J. is a magic square of palindromes,
with S = 666. It is an antimagic gnomon because the C’s = 989, 878, 898,
and 787.
K. Rotating the outer numbers results
in a gnomon square with C = 888.
L. The Durer (1514) magic square is an
example of an order4 Gnomonmagic square because the four cells in each
quadrant sum to 34.This is true for all
432 order4 squares of Groups I to
VIA.
[1] Charles W. Trigg, Third Order GnomonMagic
Squares, Journal of Recreational Mathematics, 18:1, 198586, pp 2536
Charles W. Trigg, Third Order GnomonMagic Squares with Distinct
Primes, Journal of Recreational Mathematics, 19:1, 1987, pp 12
Charles W. Trigg, A Remarkable Group of GnomonAntimagic Squares,
Journal of Recreational Mathematics, 19:3, 1987, pp 164168
[2] Annemarie Schimmel, The Mystery of Numbers, Oxford University
Press,1991, 0195089197
Pickover's Puzzle
On the cover of the now defunct REC [1],
Clifford Pickover posed as a puzzle, an unusual number square.
You have 25 squares in a 5x5 array. They contain 12 ones, a single two,
and the rest zeros. Rearrange the digits so the sum of each row, column,
and the two
main diagonals are positive and even. See
a. for the puzzle and
b. for Pickover’s solution.
The following issue of REC contained a number of
additional solutions, including c.
from Dennis Sulllivan and a similar symmetrical one
d. from Harry J. Smith.
An example of an unsymmetrical one
is e., from Don Kirk, who reported
that there is at least one solution with the 2 in an given position! Dr.
Robert Stong
reported that this problem could be solved by starting with
an order 5 magic square. First replace every odd entry with a zero, then
every even entry with a one.
Finally, replace one of the zeros with a
two.
[1] Recreational & Educational Computing, Dr. Michael Ecker, Volume 12,
Number 3 (and 4) January (and March) 1998.
Knecht's
Topographical Models
Imagine a magic
square where each cell is a square tower equal to the height of the cell’s
numerical value. Craig Knecht refers to this model as a topographical
magic square. He has explored this type of magic square for the feature of
how much water would be retained in the central part of the square.
Topographical
squares and some results of his discoveries are displayed on
his new web site at
http://www.knechtmagicsquare.paulscomputing.com/
Figure A.
shows top and side views of a 3 x 3 model (not magic) where water
could be poured into the center cell which would retain 1 unit
before pouring over the cell with height equal to 1.
Of the 880 order4 magic squares he
found that 137 would retain no water.
B. shows an order4 that
retains 7 units of water.
Craig shows topographical squares of
other orders on his site. 

On July
30, 2008, I received this more complex topographical magic square from him.
C. contains a 'lake' with an 'island' in it.
On his
site Craig also demonstrates mass in regards to magic squares and some work with
paths.
He also has a page about the Formula One constraint Logic programming language.
Craig Knecht,
Walter Trump, Francis Gaspalou, and Peter Loly investigated moment of inertia of
magic squares.
See updated information on topographical magic squares on my
2009 and
2010 Update pages.
Steimel's
Missing Digit Squares
Wilfred Steimel of Germany specializes in unusual magic squares.
Here are two of his nonnormal magic squares in which some digits
are not used. I received these by email on May 31, 2008.
a. is pandiagonal magic but
contains no 0. 6, 7, 8, or 9 and so is not normal. S = 165
b. is nonnormal, simple,
concentric, and contains no digit 9. S for orders 3, 5, 7 and 9 are
183, 305, 427, and 549. The sum of the 9 colored cells in each of
the 3 sets equals the S_{9} value, 549. 

Boyer's Multimagic
Recently
Christian Boyer added a new Update page to his excellent
Multimagic
site. I leave it to you to explore these new additions.
Here I show
two representative squares from elsewhere on his site.
Since 2007,
Lee Morgenstern has constructed several orders 4, 5, and 6
semimagic addmultiply magic squares, of which this 6x6 one is
perhaps the best.
He proved
that orders 3 and 4 addmultiply magic squares are impossible. Is an
order 5, 6, or 7 possible?
There are no known addmultiply magic squares smaller then orders 8
and 9. The first orders 8 and 9 were constructed by W. W. Horner in
1955 and 1952 respectively.
S = 360 P =
8491392000
All rows, columns and the 2 diagonals sum correctly
All rows and columns produce the same product, but the 2 diagonals
are incorrect
To the right
is a 6x6 magic square of cubes constructed in 2008 by Lee
Morgenstern. It uses all the integers from 18 to 18 (except 0). S =
0.
It is unknown
if regular magic squares of cubes are possible for orders 48
(order3 has been proved impossible).
Christian
Boyer constructed the first known orders 9, 10 and 11 normal magic
squares of cubes in late 2006. 
AddMultiply
Square
Square
of Cubes 
Gutierrez's
Semiassociated
Edward Gutierrez has a
new site with the long name
ODD MAGIC SQUARES USING TEN NEW ALGORITHMS AND PARITY
THE WHEEL, LOUBÈRE AND MÉZIRIAC METHODS 

This site introduces two new methods used for the construction of
semiassociated odd magic squares. The method consists of pairing numbers
in complementary
fashion, partitioning these complementary pairs into
groups and placing these groups into a square in a certain order using
parity as an aid.
This example is taken from his New Modified Wheel and De La Loubère
Methods
Some of the complementary pairs are symmetrical around the center cell,
but some pairs are not.
Loly's Gray Code
Square
In one of his recent papers [1] Peter Loly [2] discusses a large
variety of magic square and hypercube issues. Included is mention of an
unusual pandiagonal
nonmagic number square. If the decimal integers
(015) are converted to a special binary number system called Gray code.
binary, adjacent cell numbers, in
both the horizontal and vertical
direction change by just 1 digit. This example is one of 48 such squares
out of a total of 3,465,216 order4 pandiagonal nonmagic
squares.
Gray code Definition: An ordering of 2n binary numbers such that
only one bit changes from one entry to the next.
An nbit Gray code corresponds to a Hamiltonian cycle on an ndimensional
hypercube.
Hamiltonian cycle Definition: A path through a graph that starts
and ends at the same vertex and includes every
other vertex exactly once.
The gray code is sometimes referred to as reflected binary, because the
first eight values compare with those of the last 8 values (of regular
binary), but in reverse
order.
This table is presented for the convenience of those not too familiar
with the binary (radix 2) number system, or with the Gray code.
Decimal 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
Binary 
0000 
0001 
0010 
0011 
0100 
0101 
0110 
0111 
1000 
1001 
1010 
1011 
1100 
1101 
1110 
1111 
Gray 
0000 
0001 
0011 
0010 
0110 
0111 
0101 
0100 
1100 
1101 
1111 
1110 
1010 
1011 
1001 
1000 
Decimal (Gray) 
0 
1 
3 
2 
6 
7 
5 
4 
12 
13 
15 
14 
10 
11 
9 
8 
 A normal decimal number square.
 A Gray code decimal equivalent number square.
 A gray scale pandiagonal number square with the following
features:
adjacent cell numbers in both horizontal and vertical directions
change by only 1 digit
adjacent cell numbers in both diagonal directions change by 2
digits
adjacent cell numbers around any 2x2 square change by only 1
digit (compactlike)
any 2 cells spaced m/2 along any pandiagonal are complements
(1’s and 0s are reversed) (completelike)

The 2digit numbers in
the top row appear as least significant digits in each column.
The 2digit numbers in the left column appear as most significant
digits in each row.
Notice that the square is constructed simply by filling the rows in
alternate directions, sequentially from the string of Gray code
values. 
[1] Peter D. Loly, Franklin Squares: A Chapter in the Scientific
Studies of Magical Squares, Complex Systems, 17 (2007) 143–161
[2] Some of his other papers on magic squares are at
http://home.cc.umanitoba.ca/~loly/
Gaspalou Enumerations
Has a site at
http://www.gaspalou.fr/magicsquares/ with much material on the
methods and tools for enumeration of magic squares.
Among other results he confirmed the total (275,305,224) for order5
squares. This value was first determined by Richard Schroeppel in 1973,
then independently
confirmed by Walter Trump.
Campbell's
Pan antimagic squares
Neil Campbell has discovered a new class of antimagic squares with
features similar to the pandiagonal magic square.
A normal antimagic square has sums for the orthogonal lines and the
main diagonals that must be unique and consecutive. (if the sums are not
consecutive,
the figure is a heterosquare.)
By changing the above definition slightly, it is possible to construct
a square with the above features plus all the broken diagonals also
summing correctly.
This means that the square can then by transformed to
another pandiagonal (nasik) antimagic square by moving a line from one
side to the opposite side.
The slight change in the definition is:
A panantimagic square must contain all numbers from 0 to m^{2}
and it must have pansums that can be mapped to a set bounded by the
magic constant plus or minus half the set size. The set must have
all consecutive numbers except skipping the magic sum.The order4
example has 32 sums in consecutive order. Sums range from 14 to 46
skipping the magic constant 30.
I have a page on antimagic and heteromagic squares
here. 

Neil's father Dwane has a web site that discusses Nasik (Perfect) magic
hypercubes and has some hypercube generator programs available for
downloading.
He discusses at some length, theory of construction and
methods of testing nasik hypercubes. His site is at
http://magictesseract.com.
Dürer
Magic Square

Albrecht Durer (14711528 was a skilled artist and engraver in Germany.
On of his most famous works, "Melencolia" was produced in 1514. A prominent feature of this
work was
an order 4 magic square.
This square is notable for it’s early date of construction, but
also because it has a great many features aside from the standard rows,
columns and main diagonals summing correctly.
It is 1 of the 48 associated semipandiagonal order 4 squares of
Dudeney’s Group III.
All have the same features. There are a total of 880 distinct magic squares
of order4.
All were enumerated by Bernard Frénicle de Bessy (16051675) and
published (posthumously) in 1693.
[1] 

I recently ran across a paper by William H. Press, Did Durer
Intentionally Show Only His
SecondBest Magic Square?
[2] It shows a variation.
By simply exchanging the second and
third rows the the magnitude of the numbers in the four columns
alternately descend and ascend.
Patterns that sum to 34. The 5 rows, 5 columns, 2 main
diagonals, 2 pairs of broken diagonals plus the 8 other patterns. 
16 
3 
2 
13 
9 
6 
7 
12 
5 
10 
11 
8 
4 
15 
14 
1 
W. Press alternate square

This section was written and posted on March 10, 2012 when it was
brought to my attention that my website made very little mention of this most
famous square.
Thanks, Paul Michelet.
[1]
http://magicsquares.net/order4list.htm
[2]
www.nr.com/whp/notes/DurerSquare.pdf
More on Perfect
Pandiagonal magic squares have also been called perfect. But with magic
cubes it has led to much confusion.
Who first introduced this term? A fairly short search uncovered some early
users.
F. A. P. Barnard, on the first page of his paper says
Magic squares are
distinguished into two classes, ordinary and perfect,...'
If the sums of the numbers in the broken diagonals, equal in every case,
the
sums found in the vertical and horizontal rows, the square is perfect.[1]
Emory McClintock, On the 2^{nd} page of his paper Most Perfect
Forms of Magic Squares, he states
By reason of these remarkable
properties pandiagonal squares have received from
some writers, beginning with La Hire, the name ‘perfect”, and with others,
beginning with Lucas, the name “diabolic”.[2]
The term Nasik, we know, was introduced by A. H. Frost in 1866
[3] for pandiagonal magic squares
and pandiagonallike magic cubes. It was refined by
C. Planck in
1905 [4][5] to cover all dimensions
of hypercubes where all possible lines sum correctly!
See my Nasik
article for reasons why we should replace the term perfect
with the term nasik!
[1] F. A. P. Barnard, Theory of Magic Squares and Cubes, Nat. Academy
of Sciences, Vol. IV, Sixth Memoir, 1888, pp 207270
[2] Emory McClintock, On the Most Perfect Forms of Magic Squares, with
Methods for their Production, American Journal of Mathematics, 1897, 19,
pp 99120.
[3] A. H. Frost, Invention of Magic Cubes, Quarterly Journal of
Mathematics, 7, 1866, pages 92102. See page 99, para. 23 and page 100,
para. 26.
[4] C. Planck, The Theory of Path Nasiks, Printed for private circulation
by A. J. Lawrence, Printer, Rugby (England),1905 (Available from The
University Library, Cambridge).
[5] W. S. Andrews, Magic Squares and Cubes. Open Court Publ.,1917. Pages
365,366, by Dr. C. Planck.
Republished by Dover Publ., 1960 (no ISBN); Dover Publ., 2000,
0486206580; Cosimo Classics, Inc., 2004, 1596050373
