Problems & Puzzles: Puzzles

Puzzle 165.  Bye Mr. Suzuki

" I hope to see you again sometime and somewhere
in another site of magic squares ",
Mr. Mutsumi Suzuki's closing Web-page

While not all the magic squares are such that its reversible ones are also magic squares, some of them really remain magic.

My friend Jaime Ayala not only introduced the subject but also produced the least of them

 32 14 53 23 41 35 54 33 12 reversible à 45 33 21 13 52 34 31 25 43

As you may suppose immediately I wanted to know if there are examples of prime-magic squares that:

a) their reversible are also magic
b) their reversible are also magic and prime

I offer an example-solution for a):

 Prime magic square Magic reversible but not-all-prime 52223 10103 34313 32225 30101 31343 14303 32213 50123 30341 31223 32105 30113 54323 12203 31103 32345 30221

* as usual, primes are in blue

Question:

Can you find a prime-magic square such that its reversible one is also prime-magic?

Solution:

Jean-Claude Rosa wrote (24/1/2002):

I have found  several solutions like you for question a) , by example:

A)  17539     13729    15679         B)  93571     92731     97651
13789     15649    17509               98731     94651     90571
15619     17569    13759               91651     96571     95731

(numbers in bold letters are prime )

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This solution from Rosa is very important (at least for me) because I was thinking that the central number had certain limiting value that... but forget it... all this I was thinking is evidently wrong...

***

J C Rosa smallest solution is:

7789     7039     7549              9877     9307     9457
7219     7459     7699              9127     9547     9967
7369     7879     7129              9637     9787     9217

and an improved solution (6 primes in the reverse magic square) from himself:

195479    111869    159389           974591   968111     983951
119489    155579    191669           984911    975551    966191
151769    199289    115679
967151    982991    976511

***

I  (C.R.) made again some search eliminating certain artificial conditions that I added for free previously to my code, and I have gotten one smaller solution with 6 primes in the reversible solution, than the provided by J.C. Rosa:

13469 11279 12689          96431 97211 98621
11699 12479 13259  --> 99611
97421 95231
12269 13679 11489          96221
97631 98411

(primes in bold)

***

J.C. Rosa found (27/1/02)  one solution with 7 primes:

167621 119291 155741          126761 192911 147551
135671 147551 159431 -->   176531 155741 134951
139361 175811 127481
163931 118571 184721

(primes in bold)

Later he reported other two, but not the following found by me, also with 7 primes:

175873 112213 175543       378571 312211 345571
154213 154543 154873       312451 345451
378451
133543 196873 133213
345331 378691 312331

***

J. C. Rosa has gotten (9/2/02) an almost complete solution, one in which the reversed matrix has only one composite number!

1335211    1043761    1324621       1125331    1673401    1264231
1223941    1234531    1245121       1493221    1354321    1215421
1144441    1425301    1133851       1444411    1035241    1583311

(primes in bold)

***

The game is over and the winner is... Jean Claude Rosa. The 3/3/2002 he found the 9 reversible, distinct and non-palindromic primes 3x3 magic square:

 18544613 10175573 14646743 31644581   37557101   34764641 10557773 14455643 18353513 37775501   34655441   31535381 14264543 18735713 10366673 34546241   31753781   37666301

A big clap for this!

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