[ January 5, 2001 ]
A Palindromic & 'Sophie Germain' pair of magic 3x3 squares
A beautiful submission from Jaime Ayala (email)

252	171	363		505	343	727
373	262	151	2(n)+1	747	525	303
161	353	272		323	707	545

The Magic Sums are resp. 786 and 1575.

No problem if you're unfamiliar with Sophie Germain numbers.
WONplate 29 has lots of links!
Jaime Ayala submitted other magic squares in the past.
Visit for instance WONplate 14 or palprim3.htm

Carlos Rivera (site) poses an extra puzzle question :
Are there examples of these magic squares using palindromic prime numbers ?

We can also investigate magic squares of the 2nd SG type 2(n)–1.
Palindromic solutions seems not possible in this case...
What about solutions with mere primes ?

Carlos Rivera wrote me [ January 21, 2001 ] that the statement
"There are no couples of palindromes of the SG 2nd type" is false.
The phrase is only correct if Pal1 & Pal2 are both primes.
As a matter of fact any & only strings of 6's provide composite palindromes Pal1
such that Pal2 is prime (2*Pal1 – 1 = Pal2)
2*6 – 1 = 11
2*66 – 1 = 131
...
2*(6)_k – 1 = 1(3)_k–11
The palindromes Pal2, of the kind 1(3)_k–11, are primes for
k = 1, 2, 4, 94, 160, 360, ...

Regarding the magic squares of the SG 2nd type (order) using only primes,
please see my Puzzle 81, where the topic has been fully investigated.

From Jaimito (Jaime Ayala's son) came the following apparently
innocent question to his father [ January 22, 2001 ]
"Can you now produce a palindromic magic square 3(n)+1
starting from a palindromic magic square one ?"
Jaime's immediate answer was that it was not possible...
but then he almost satisfied his son's request through an extremely unexpected route.
He thereby produced a beautiful palindromic magic square that isn't 3(n)+1 cell by cell
but magic nevertheless and with a magic sum of 3(S)+3
where S is the magic sum of the first square.
The three magic squares involved use 27 distinct palindromes.

The first magic square is the original one from above but rotated 180 degrees.
The second magic square is the original second SG from above kept unaltered.
The third magic square is formed by adding the corresponding cells of the first two.
The magic sum of this third square is 2361
which is 3(S)+3 with S = 786 the magic sum of the first square !
Note that when written in hexadecimal we have
2361_{10} = 939_{16}... another threedigit palindrome !

Carlos Rivera remarks that one more time it has been confirmed that young people
have smart eyes and produce or promote unusual paths to the mind !