Type 2 Order-3 Magic Squares

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

Type 1
8 1 6
3 5 7
4 9 2
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 Fuh-Hi of China (2858-2738 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 non-consecutive numbers.


 

Type 2
8 1 7
4 5 6
3 9 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 non-consecutive 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 non-consecutive numbers.

This was as a result of his investigation of the results of Harry L. Nelson, who discovered the smallest possible Order-3 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 non-consecutive prime number magic square, and had this explanation.

Non-consecutive 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 order-3 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 e-mail from Aale de Winkel commenting on the possibility of a Type 2 order-3 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 e-mailed 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 e-mailed Aale a copy of Harry Smith's letter, mentioned above
August 1, 1999 I passed on Harry Smith's e-mail address to Aale and suggested he contact him direct to compare notes.
August 4, 1999 I received a CC of Harry's e-mail to Aale confirming  he had indeed discovered a Type 2 order-3 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
8 1 7
4 5 6
3 9 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 231 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 order-3 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 order-3 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

9

1

8

5

6

7

4

11

3


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

  1. Harry L. Nelson, A consecutive-Prime 3 x 3 Magic Square, Journal of Recreational Mathematics, 20:3, 1988, pp214-216.
  2. 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://harry-j-smith.com/hjsmithh/  (sorry. No longer available.
         REC home page is http://members.aol.com/DrMWEcker/REC.html  (No longer online)
  3. Aale de Winkel's home page is   http://www.magichypercubes.com/Encyclopedia/index.html
  4. Carlos Rivera's Prime Puzzles and Problems page is http:/www.primepuzzles.net

 

This page was originally posted August 1999
It was last updated July 23, 2011
Harvey Heinz   harveyheinz@shaw.ca
Copyright 1998-2009 by Harvey D. Heinz