August 4, 1999
Harry J. Smith confirms that Aale de Winkel has discovered a Type 2 magic square!
Type 1

This magic square is the one we are all familiar with. It is thousands of
years old, and is incorporated in the Loh_Shu, credited to FuhHi of China (28582738
B.C.). This is a normal magic square. However, the digits may also be used as position indicators for the magnitude of the numbers, when constructing a magic square using nonconsecutive numbers.


Type 2

This is NOT a magic square and no such square can be constructed using
consecutive numbers. However, the digits may be used as position indicators for the
magnitude of the numbers, when constructing a magic square using nonconsecutive numbers. In December of 1990, Harry J. Smith suggested in a letter to Dr. Michael W. Ecker, editor of REC, that it may be possible to construct a magic square of this type, using nonconsecutive numbers. This was as a result of his investigation of the results of Harry L. Nelson, who
discovered the smallest possible Order3 magic square of consecutive primes. 
Harry Nelson, in an paper in The Journal of Recreational Mathematics in August 1988, used this example of a nonconsecutive prime number magic square, and had this explanation.
Nonconsecutive primes

Here the triplets are 29, 41, 53; 59, 71, 83; 89, 101, 113. The magic sum
is 3 x 71 = 213. Like all 3 x 3 magic squares, it adheres to the pattern
with a magic sum of 3a + 9b + 3c (i.e. three times the middle term). 
See my Prime Squares page for the details on the smallest consecutive prime numbers order3 magic square. This discovery by Harry Nelson is what probably got the whole investigation going
Now for the exciting part!
During this time I was involved with another project (when not out of town on holidays) and am sorry to say didn't fully appreciate what Aale de Winkel was accomplishing! Following are the highlight. Many other messages were exchanged.
July 9, 1999  I received an email from Aale de Winkel commenting on the possibility of a Type 2 order3 magic square as mentioned on my Prime Squares page. He requested the other 20 consecutive prime sequences that Harry Nelson had discovered. 
July 23, 1999  Aale emailed Carlos Rivera (with a CC to me) with the announcement that he had posted a page on Magic Sequences and informing us that the Nelson squares seem to have 4 different magic sequences. 
July 28, 1999  I emailed Aale a copy of Harry Smith's letter, mentioned above 
August 1, 1999  I passed on Harry Smith's email address to Aale and suggested he contact him direct to compare notes. 
August 4, 1999  I received a CC of Harry's email to Aale confirming he had indeed discovered a Type 2 order3 prime number magic square! 
The 1st Type 2 consecutive prime number magic square

This magic square uses the 21st prime sequence discovered by Harry Nelson
. It consists of 3 triplets with internal steps of 30 and steps between the triplets of 6. The magic sum is 71440079091 which, of course, is not prime. It is 3 times the central number. 
Type 2

The magnitude of the numbers in the above square are arranged as per the
square on the left. So now, in hindsight, the difference between a Type 1 and a type 2 is simply the sign of the step between the three triplets. 
The 2nd Type 2 consecutive prime number magic square

This type 2 magic square uses the 22nd prime sequence discovered by
Harry Nelson . It consists of 3 triplets with internal steps of 30 and steps between the triplets of 18. I guess nobody before realized that these were Type 2 squares. 
Harry Smith arrived at his conclusions by the use of 8 equations and 9 unknowns. In fact he found the same 2 consecutive prime magic squares that Harry Nelson had found, but by specifically searching for type 1 and 2 squares based on his analysis. He extended his search only to 2^{31} and was obviously unaware of Nelson's 19th and 20th sequences.
Aale de Winkel used a method he calls magic sequences which he used to reconstruct the magic square. For example; the last magic square above may be constructed with his magic sequence {5,2,7,12}_{6}. See his web page for details.
An order3 magic square may be constructed with any set of 9 numbers as long as there are 3 sets of 3 numbers (triplets) with common difference (step) between the numbers of the 3 triplets, and there is a common (possibly different) step between the 3 triplets.
The (vertical) step between triplets is positive for Type 1 squares and negative for
type 2 squares.
The normal order3 magic square with numbers 1 to 9 simply has both steps equal to 1.
Easy type identification (with the smallest number in the middle of
the top row):
Type 1. The 3rd number, by magnitude, is in cell 1 of the middle row.
Type 2. The 3rd number, by magnitude, is in cell 1 of the bottom row.
Smallest Type 2

The triplets are: 1, 3, 5; 4, 6; 8, 7, 9, 11. (the horizontal step is 2,
the vertical step is 1). They are simply arranged in the magic square in the order of a normal type 1. This is the smallest type 2 it is possible to construct using the natural numbers. 
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Harvey Heinz harveyheinz@shaw.ca
This page last updated
October 31, 2009
Copyright © 1999, 2000 by Harvey D. Heinz