Contents
This latest
update page was inspired by several interesting items I recently received
from other magic square hobbyists.
I have taken this
opportunity to also include small interesting items I have been collecting over
recent years.
Hopefully you, the reader, will also find them interesting!
Amela Fundamental Solution
In May 2010
Miguel Angel Amela of Argentina sent me a paper where he presented
algorithms for constructing all order 4 and order 5 pandiagonal magic
squares from a fundamental square.
[1]
This was
previously demonstrated for the order5 pandiagonals by Benson and Jacoby
in 1976
[2]
by means of an algebraic square. I show this method on my
Pandiagonal 5x5 page
However, Amela
demonstrates how to do this by cell exchange transformations from any one of
these 3600 magic squares.
Rather then attempt to explain these transformations I have made his paper
available for download (with his permission).
[3]
Here I present an
interesting but unrelated footnote that he had added to his page in the final
revision.
The 3 x 3
squares are the nine ways these numbers may be arranged in an array with
all diagonals summing to 15. However, most of the rows and columns do not,
so the squares are not magic. The nine arrays are arranged to form an
order9 simple magic square. The center number of each 3 x 3 array form
one order3 simple magic square!
Order5 Pandiagonal multiplication
From
another paper
[4]
by Miguel Amela, I found this order5 multiplication magic square.
According to his paper, this is one of exactly 3600 unique
pandiagonal solutions.


Multiplicative diamond compact pandiagonal square
A few days
after posting this page, Miguel sent me
[5]
an example of an unusual order6 square. While not magic in the
conventional sense, it has it's own type of magic.
Zigzag
cells of any two adjacent lines (horizontal or vertical) produce a
constant product. An example is shown starting at cell 1 of top line in
illustration. Also, any 4 cells in a 2x2 diamond pattern, as shown in
lower right corner, also produce a constant product. Either pattern may start on any of
the 36 cells in the square (using wraparound). 4 cell products
are 1,587,600. 6 cell products are 2,000,376,000. 

[1] Amela’s paper released in
June 2010, is titled Fundamental Solution in Magic Squares of Order
Four and Five.
[2] Benson and Jacoby, New Recreations With Magic Squares, Dover
Publ. 1976, 0486232360,
[3] My download page is here.
[4] Miguel Amela, Total Compactness in Magic Squares of Order Five.
[5] email dated July 17,2004
Knecht Topographical squares
Craig Knecht
recently sent me an update on his topographical squares. The following
information is condensed from that email.
Review the subject on my last squareupdate page.
[1]
For more indepth information refer to Craig’s web site.
[2]
The following
image shows simple magic squares for orders 7, 8, and 9. Included with each
square is the author, date, and units of retention.
In each case,
this is the maximum possible for that order.
Note that for order 7, there are 2 ‘lakes’ (12 and 3 cells) and 2 ‘ponds’. For
order8 there are 2 lakes (14 and 5 cells) and 4 ponds. For order9 there are
also 2 lakes (25 and 3 cells) and 6 ponds. Coincidence?
Al
Zimmermann
[3]
ran a contest to see who would find the largest amount of retained water
for each magic square up to order28. The contest ended June 12, 2010.
Following are the first author to find the maximum for each order. Note
that these are not proven to be the maximum possible!
Order

Author 
Date

Units 

Order

Author 
Date 
Units 
6 
Wes Sampson 
Mar. 14, 2010 
192 

18 
Jarek Wroblewski 
March 23, 2010 
31,871 
7 
Hermann Jurksch 
Apr. 6, 2010 
418 

19 
Jarek Wroblewski 
March 23, 2010 
40,473 
8 
Hermann Jurksch 
Apr. 5, 2010 
797 

20 
Jarek Wroblewski 
March 24, 2010 
50,754 
9 
Walter Trump 
June 12, 2010 
1,408 

21 
Jarek Wroblewski 
March 24, 2010 
62,877 
10 
James J Youlton Jr 
Apr. 12, 2010 
2,267 

22 
Jarek Wroblewski 
March 23, 2010 
77,098 
11 
Hugo Pfoertner 
Apr. 22, 2010 
3,492 

23 
Jarek Wroblewski 
March 23, 2010 
93,623 
12 
Hermann Jurksch 
June 10, 2010 
5,185 

24 
Jarek Wroblewski 
March 24, 2010 
112,710 
13 
Walter Trump 
May 5, 2010 
7,442 

25 
Jarek Wroblewski 
March 23, 2010 
134,598 
14 
Frederic van der Plancke 
May 19, 2010 
10,397 

26 
Jarek Wroblewski 
March 24, 2010 
159,565 
15 
James J Youlton Jr 
Apr. 15, 2010 
14,154 

27 
Jarek Wroblewski 
March 23, 2010 
187,880 
16 
James J Youlton Jr 
Apr. 4, 2010 
18,887 

28 
Jarek Wroblewski 
March 24, 2010 
219,822 
17 
Jarek Wroblewski 
Mar. 23, 2010 
24,730 

100 
Craig Knecht 
August 14, 2009 
34,788,903 
It is interesting
that there are several maximum solutions for many orders. Order6 for example
had 20 different solutions submitted to the contest between March 14 and June 8,
2010 with 192 units of retention.
Order 18 is interesting because 2 solutions were submitted, showing 31,871
units. The one found by Walter Trump has 199 cells and the one found by Jarek
Wroblewski requires 200 cells! Since the contest ended, Walter Trump has already
bettered this order18 record to 31,872!
As a point of
interest: Craig sent me an order 100 magic square on Aug. 14, 2009. Magic sum
was 500,050. Units of retention totalled 34,788,903!
[1] Information previously posted
on my site is here
and here.
[2] Craig Knecht’s site is at
http://www.knechtmagicsquare.paulscomputing.com/
[3] Al Zimmermann's Programming Contests at http://www.azspcs.net/Contest/MagicWater (Sorry.
No longer available.)
Campbell complete order8
I received
the following order8 magic square (a.) from Dwane Campbell in November
2009.
The square is
pandiagonal, and is composed of 2 by 2 blocks of cells, with each of these
containing 4 consecutive numbers.
The order4 square
(b.) is also pandiagonal magic and consists of the sums of each of the 16
blocks.
As an additional
feature, the identical corner of each 2x2 block can be combined to make 4
additional order4 pandiagonal magic squares. One of these (also pandiagonal) is
shown (c.).
The above
squares are all ^{2}complete_{2} because 2 integers placed
m/2 along a diagonal sum to the same value (S/4 for the order8 and
S/2 for the order4 squares).
The order8 square
is not compact even though it consists of 2x2 blocks, because the blocks do not
sum to the same value. However, corners of 3x3 arrays sum correctly so the
order8 is compact_3. (It is also compact_5 as a result of being complete and
compact_7 as a result of being compact_3.)
More
information on complete and compact is
here.
Gus Arrington’s model square
Gus
Arrington of Boyd, Texas sent me the order9 magic squares shown in the
image as Squares a. and c. Square c. is a normal simple magic square with
integers from 1 to 81. Square a. uses integers from 82 to 162. He also
sent some images of a wooden ball he constructed. Following is his
description of the model.
The above
squares are placed back to back on the ball, so that each circuit round
the ball passes through 18 numbers that sum to 1467. Each ring on the ball
will rotate independently, and as long as the main diagonal numbers on all
rings are aligned, the rows, columns, and main diagonals will sum
correctly.
I have
reproduced Square b. from the images of the model, and indicated the
rings. It is square a. with the 2nd and 3rd rings from the center rotated.
Alex de Wit Squares
Several
years ago, Alex de Wit sent me a number of unusual magic squares. I am
taking this opportunity to finally include some of them on my site.
Upside
down
This
magic square is composed of Roman Numerals. When rotated 180 degrees
it forms another magic square with a different arrangement of the
same integers. All rows and columns, the two diagonals, and the 4
corners plus the center cell sum to XLIV (or 44). 
T square
This
simple magic square consists of numbers all starting with the letter
‘T’.

Broken
square
Although several cells are missing, all rows, columns, main
diagonals, the four corners, and the center 2x2 add up to 32. 
Magic
rectangle
In
this associated magic rectangle, 3 rows, 5 columns, 6 diagonals, the
4 corner cells, and the corners of the central 3x3 all sum to 0! 
NexttoMain Diagonals
This is a
simple magic square.
However, it
has the feature that the two 7 cell diagonals parallel to each main
diagonal also sum to S


Add Multiply Orders 6 and 4
his
order 6 simple magic square has all rows, columns, and main
diagonals summing to 1355. Embedded in it is an order4 magic
multiplication square with all 8 lines producing a product of
401,393,664.
If the digits of each integer in the 4x4 square are reversed (408
becomes 804, etc), the square remains magic, with a product of
4,723,906,824.
It was
constructed before 1966
[1]
by Ronald B. Edwards of Rochester , New York 

[1] Joseph S.
Madachy, Mathematics on Vacation, 1968, 17 147099 0, page 90. ©
Joseph S. Madachy 1966
Prime Number
Talisman Heterosquare
This
order5 heterosquare consisting of consecutive prime numbers from
4673 to 4909 was constructed by Enoch Haga in 2004.
The 5
rows, 5 columns, and 2 main diagonals all sum to different values
from 23565 to 24349. It also has the following features! 

This
is also a talisman square because the minimum difference between adjacent cells
(in this case) are greater then 17.
All 16 2x2
arrays (wraparound doesn’t work here because of duplicate pairings) have unique
minimum differences between the 6 cell pairs of each array. For example, the
minimum of six differences in the top left array is 47294673 = 56. The
differences for the other similar arrays are 60, 54, 62, 42, 18, 30, 26, 58, 34,
48, 44, 28, 24, 38, and 32. The minimum difference for the entire order5 square
is the smallest of these values. i.e. 18.
This square is
an elaboration of a type of square array investigated and named by Sidney
Kravitz. A talisman square (not necessarily magic) must have a minimum
difference between adjacent cells (horizontal, vertical, and diagonal) greater
then some specified amount. See square b with it's minimum difference of 3.
[1]
[1] J. S. Madachy, Madachy’s
Mathematical Recreations, Dover Publ. , 1979, 0486237621 pp.110113
Magic  Antimagic Combined
Some
years ago Carlos Rivera
[1]
suggested a problem involving order5 magic or antimagic squares
containing an order3 antimagic or magic square. Below are shown several
of the results he obtained.

Kurchan square
The
3x3 square is magic with S = 39. The 5x5 is antimagic with S = consecutive
numbers 59 to 70.
This square uses consecutive numbers from 1 to 25 so it is ‘normal’.
It is a ‘bordered’
[2]
square because the 9 central numbers (of the range) are in the center
square.
Rosa
square
The 3x3 square is antimagic with S = 227 to 241. 5 of these 8 sums are
prime numbers.
Technically this square is a heterosquare because the sums are not
consecutive numbers. However, they are consecutive odd numbers, and
the square consists of all odd numbers!
The 5x5 square is magic, with S = 389 (which is prime). It consists of 25
of the first 36 prime numbers. 
[1] See Carlos
Rivera’s
http://www.primepuzzles.net/puzzles/puzz_263.htm
[2] See my antimagic squares page for more on
antimagic and heterosquares.
Miscellaneous Tidbits
As mentioned
at the start, not all items on this page are recent. Here are a variety I
have collected over the years.
 Two order4
magic squares, one the reverse of the other.
 An order7
magic square uses the 16 primes between 1 and 49 to form the number ‘19’.
[1]
 An order3
prime number magic square that sums to 15
 An order3
magic square (so called) consisting of the first 9 integers of the Fibonacci
series.
The sum of the products of the 3 rows equal the sum of the products of the 3
columns.
 Order5 with
primes arranged as a T. I previously posted a square like this constructed by
H. E. Dudeney
[2]
 An order5
with the top row consisting of the first 9 digits of pi.
[3]
 An order5
multiply magic square with magic product = 1
[4]
 An order4
pandiagonal magic square consisting of twin primes.
[5]
[1] Anurag Sahay.
Email of May 2, 2005
[2] Dudeney's T square is on my
Prime Squares page
[3] Benson and Jacoby, New Recreations With Magic Squares, Dover
Publ. 1976, 0486232360, p. 47
[4] Supplied by Ed Shineman Jr. He reports it was shown at the Believe it
or Not Odditorium at the Chicago World Fair 1933
[5] Sabastião A. DiSilva email of March 26, 2005
