[ *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–1}1

The palindromes Pal2, of the kind 1(3)_{k–1}1, 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.

272 | 353 | 161 | 505 | 343 | 727 | 777 | 696 | 888 |

151 | 262 | 373 | 747 | 525 | 303 | 898 | 787 | 676 |

363 | 171 | 252 | 323 | 707 | 545 | 686 | 878 | 797 |

N2 = N, 180° rotated | *SG* = 2(n)+1 | N2 + *SG* |

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 !