- **Puzzles**
(*https://www.mersenneforum.org/forumdisplay.php?f=18*)

- - **Square of Primes**
(*https://www.mersenneforum.org/showthread.php?t=10251*)

Square of PrimesConstruct a 5 x 5 square containing distinct primes
such that each row, column and diagonal sums to a distinct prime. |

[QUOTE=davar55;132471]Construct a 5 x 5 square containing distinct primes
such that each row, column and diagonal sums to a distinct prime.[/QUOTE] Is it a magic square where every sum is the same? |

A magic square of primes (where every sum is the same) is
solved elsewhere (although it would be a perfectly good puzzle to re-solve). Here every sum is a different prime. |

The original problem was perhaps too computationally simple
to be interesting. The following additional condition adds an iota of complexity: The 25 distinct primes in the square should be the first 25 odd primes {3,5,7,...,97,101}. (I have a solution which wasn't hard to find by trial and error, so there must be many solutions; but plain brute force on the 25! such possible squares is obviously too computationally costly.) |

Here's one:[CODE][SPOILER] 239
/ / / 3, 5, 7, 11, 17 --- 43 13, 19, 29, 23, 43 --- 127 31, 67, 61, 47, 71 --- 277 53, 59, 41, 73, 37 --- 263 79, 83, 89, 97, 101 --- 443 | | | | | \ | | | | | \ | | | | | \ 179 233 227 251 269 257[/SPOILER][/CODE] |

There is an error in your calculations : the last row total is of by 6. But the right number is prime so the solution stands :-)
Jacob |

Hm, I think I worked the total out right, but wrote it down wrong.
It wasn't just a fluke, honest! ;) |

Does the sequence
a(n) = number of square matrices containing the first (2n+1)x(2n+1) odd primes, such that row, column and diagonal sums are distinct primes exist on OEIS ? |

It would appear that there are an awful lot of these out there, so perhaps the challange should be to find a square with the lowest standard deviation of column/row/diagonal totals.
I'll start the ball rolling with a slightly modified version of the last square I posted, with an s.d. of 84.51:[code][SPOILER] 239 / / / 3, 5, 7, 11, 17 --- 43 13, 19, 29, 23, 43 --- 127 31, 67, 61, 47, 71 --- 277 53, 59, 89, 73, 37 --- 311 79, 83, 41, 97, 101 --- 401 | | | | | \ | | | | | \ | | | | | \ 179 233 227 251 269 257[/SPOILER][/code] |

Here's another solution:
041 005 007 071 003 013 023 029 031 067 059 053 043 047 037 019 011 089 061 017 079 101 083 097 073 Rows: 127,163,239,197,433 Columns: 211,193,251,307,197 Diagonals: 241,167 (Standard deviation: 76.7) An alternative measure is simply mini-max: minimize the largest sum. By that measure, lavalamp's solution is a better one. |

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