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.
Note:
Credits and links appear at the bottom of the page. 
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
101 
29 
83 
53 
71 
89 
59 
113 
41 
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
a + 5b + 2c 
a 
a + 4b + c 
a + 2b 
a + 3b + c 
a + 4b + 2c 
a + 2b + c 
a + 6b +2c 
a + b 
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 actual Type 2 magic square
The 1st Type 2 consecutive prime number magic
square
23813359751 
23813359613 
23813359727 
23813359673 
23813359697 
23813359721 
23813359667 
23813359781 
23813359643 

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
49285771793 
49285771679 
49285771781 
49285771739 
49285771751 
49285771763 
49285771721 
49285771823 
49285771709 

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 a
different type of squares. 
Conclusion
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.

Credits, References and Links
 Harry L. Nelson, A consecutivePrime 3 x 3 Magic Square, Journal of
Recreational Mathematics, 20:3, 1988, pp214216.
 Harry J. Smith, Letter to Dr. Michael W. Ecker, editor of Recreational And
Educational Computing, dated Dec. 8, 1990
It appeared in Farrago IX, Disk 4.
Harry Smith's home page is http://harryjsmith.com/hjsmithh/
(sorry. No longer available.
REC home page is http://members.aol.com/DrMWEcker/REC.html (No
longer online)
 Aale de Winkel's home page is
http://www.magichypercubes.com/Encyclopedia/index.html
 Carlos Rivera's Prime Puzzles and Problems page is
http:/www.primepuzzles.net
