S U R F
C A L M
E X P O
T I N Y

Each of the above letters represents a positive integer smaller than 200. In some order they represent an arithmetic sequence. Overall, the grid as presented is a magic square, with each row, each column and each of the two large diagonals adding to the same value.

Also, each of the letters in P-R-I-M-E-S represents a prime number, and in themselves they also form an ascending arithmetic sequence, in the order P-R-I-M-E-S.

What is this magic square?

At the moment I am thwarted.

I am fully aware that a solution appears to rest upon this 'latin' square which fulfills the structural attributes:
  1  14  15  4        S  U  R  F
 12  9    6   7        C  A  L  M
  5   8   11 10       E  X  P  O
 16  3    2  13       T   I  N  Y

Unfortunately what is presented will not work as the numerals corresponding to the letters are not in an ascending AP sequence of "P-R-I-M-E-S".

It therefore seems apparent that I need to consider:
a. flipping around the major diagonals,
b. moving rows and columns (2 at a time) to preserve the "34" structure, and
c. swapping each pair of diagonally opposite quadrants where the "34" structure is preserved.
Lastly, (mirroring/reflections are addressed above) 90º rotations, if they haven't already been addressed in the a, b and c above also warrant thought.


Currently it seems that the AP must have a constant of 2, and it could be odd or even at this point.

Then?
Having derived a satisfactory AP for "P-R-I-M-E-S" those values need to be used to generate tables of values which are ±1 from multiples of 6 and 12 but below 200.  The requirements of "P-R-I-M-E-S" must match that selection. 

Having determined values for "P-R-I-M-E-S" I assume that I'd require a computer to determine the 6! ways to concatenate the 6 values to determine 'primacy'.

Edited on August 5, 2011, 9:47 am

Posted by brianjn on 2011-08-05 06:29:44