Magic Square Update-2010



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 Knecht Topographical squares Campbell complete order-8
Gus Arrington’s model square Alex de Wit Squares Add Multiply Orders 6 and 4
Talisman  Heterosquare Magic - Antimagic Combined Miscellaneous Tidbits

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 order-5 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).

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 order-9 simple magic square. The center number of each 3 x 3 array form one order-3 simple magic square!

Order-5 Pandiagonal multiplication

From another paper [4] by Miguel Amela, I found this order-5 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 order-6 square. While not magic in the conventional sense, it has it's own type of magic.

Zig-zag 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 wrap-around).

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, 0-486-23236-0,
[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 square-update page.
[1] For more in-depth 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 order-8 there are 2 lakes (14 and 5 cells) and 4 ponds. For order-9 there are also 2 lakes (25 and 3 cells) and 6 ponds. Co-incidence?

Al Zimmermann [3] ran a contest to see who would find the largest amount of retained water for each magic square up to order-28. 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! 











Wes Sampson

Mar. 14, 2010




Jarek Wroblewski  

March 23, 2010



Hermann Jurksch

Apr. 6, 2010




Jarek Wroblewski  

March 23, 2010



Hermann Jurksch

Apr. 5, 2010




Jarek Wroblewski  

March 24, 2010



Walter Trump

June 12, 2010




Jarek Wroblewski  

March 24, 2010



James J Youlton Jr 

Apr. 12, 2010




Jarek Wroblewski  

March 23, 2010



Hugo Pfoertner

Apr. 22, 2010




Jarek Wroblewski  

March 23, 2010



Hermann Jurksch

June 10, 2010




Jarek Wroblewski  

March 24, 2010



Walter Trump

May 5, 2010




Jarek Wroblewski  

March 23, 2010



Frederic van der  Plancke  

May 19, 2010




Jarek Wroblewski  

March 24, 2010



James J Youlton Jr  

Apr. 15, 2010




Jarek Wroblewski  

March 23, 2010



James J Youlton Jr  

Apr. 4, 2010




Jarek Wroblewski  

March 24, 2010



Jarek Wroblewski  

Mar. 23, 2010




Craig Knecht

August 14, 2009


It is interesting that there are several maximum solutions for many orders. Order-6 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 order-18 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
[3] Al Zimmermann's Programming Contests at (Sorry. No longer available.)

Campbell complete order-8

I received the following order-8 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 order-4 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 order-4 pandiagonal magic squares. One of these (also pandiagonal) is shown  (c.).

The above squares are all 2complete2 because 2 integers placed m/2 along a diagonal sum to the same value (S/4 for the order-8 and S/2 for the order-4 squares).

The order-8 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 order-8 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 order-9 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! 

Next-to-Main 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 order-4 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 order-5 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!

  • All 25 2x2 arrays (including wrap-around) sum to different values between 18912 and 19386.

  • All 25 3x3 arrays (including wrap-around) sum to different values between 42747 and 43477.

  • All 25 4x4 arrays (including wrap-around) sum to different values between 76122 and 77090.

This is also a talisman square because the minimum difference between adjacent cells (in this case) are greater then 17.

All 16 2x2 arrays (wrap-around 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 4729-4673 = 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 order-5 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, 0-486-23762-1 pp.110-113

Magic - Antimagic Combined

Some years ago Carlos Rivera [1] suggested a problem involving order-5 magic or antimagic squares containing an order-3 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 anti-magic 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
[2] See my anti-magic 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.

  1. Two order-4 magic squares, one the reverse of the other.
  2. An order-7 magic square uses the 16 primes between 1 and 49 to form the number ‘19’. [1]
  3. An order-3 prime number magic square that sums to 15
  4. An order-3 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.
  5. Order-5 with primes arranged as a T. I previously posted a square like this constructed by H. E. Dudeney [2]
  6. An order-5 with the top row consisting of  the first 9 digits of pi. [3]
  7. An order-5 multiply magic square with magic product = 1 [4]
  8. An order-4 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, 0-486-23236-0, 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

This page originally posted July 2010
It was last updated February 24, 2011
Harvey Heinz
Copyright © 2009 by Harvey D. Heinz