Problems & Puzzles: Puzzles
Puzzle 4.- Prime - magical squares
Jaime Ayala and myself asked each other very recently for a type of magic squares with the peculiarity that each of the rows, columns and major diagonals should be distinct, reversible and non-palindrome primes. We called this kind of matrixes "prime-magical squares"
Our search concluded that :
by the way, we conjecture that this is the only solution available for this case.
.24 primes 5 digits each, inside
.28 primes 6 digits each, inside
We obtained the before shown matrixes observing that the "wall" of the matrixes should not contain even digits. Then we choose appropriate matching 5 or 6 digits reversible non-palindrome primes for the wall and started looking for the inner digits using a code that implement a combination of trial & error and a systematic "optimizing" of the matrix procedure.
In this case are asking if you devise a general and non trial-error procedure for obtaining this kind of solutions.
What about a genetic approach ?
Harvey Heinz (see my Link # 20 to his very interesting Web pages) is reporting that two other persons worked several years ago with this "prime magical squares". They called this matrixes "perfect prime squares".
The first one who worked on this subject is Les Card and published his results as "Patterns in Primes" at the "Journal of Recreational Mathematics, 1:2, pp 93-99, April 1968. He found the same 4x4 we found 30 years later ! ! ! and other 3 matrixes 5x5 totally different to ours. This are the Les Card ones :
13933 11731 99139
Later, Charles Trigg reproduced the Les Card results and added some other 4x4 matrixes in the "Journal of Recreational Mathematics", vol. 17, # 2, 1984-1985.
But none of the added matrixes have 20 primes, just 18 because he is using one reversible primes in two rows and/or column different, breaking the condition of our puzzle that rules that you should use only "distinct reversible and non-palindrome primes"
For example, in the following square
the first column and the last row are the same reversible prime.
The best new for us is that neither Trigg nor Les Card found any of 6x6 order prime magical square ! . while we have found last March two of them Excellent news! . and many thanks to Harvey.
At 14/05/99 Wilfred Whiteside informed that he has calculated all the solutions for the 5x5 case and found that there are 253,688 different matrixes containing 24 unique primes. He has stored all these solutions in 6 text files available on request.
In a separate e-mail but the same day, and for the 4x4 matrixes, he also wrote: "I just reran my program for all possible 4x4 solutions ...The only solution after symmetry elimination was your solution...A total of 4653 arrays were analyzed, yielding 4 equivalent versions of your solution".
After the Whiteside's work we can say that the 4x4 and 5x5 matrixes have been exhaustively investigated and solved, remaining only to know how many different solutions exists for the 6x6 case. But the task seem to be unaffordable. In his own words, Whiteside says "I definitely will not try to analyze all possible 6x6 solutions since there are probably several million sets of "walls" and for each of these, there are probably many tens of millions of ways to fill them in. It would not surprise me to find out that there are a few billion solutions to the 6x6 case."
Nevertheless, one day after Whiteside sent to me some "exotic" solutions for the 6x6 case. I have copied only a few of them:
Some Exotic Prime-Magical Matrixes
- 6x6 Case, with only 0,2,4 allowed in the interior (only
3 solutions exist):
Now you know it... Whiteside has the key of this puzzle....
"... here are some details from my 'recipe'