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Old 2008-04-30, 18:12   #1
davar55
 
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Default Square of Primes

Construct a 5 x 5 square containing distinct primes
such that each row, column and diagonal sums to a distinct prime.
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Old 2008-04-30, 22:35   #2
petrw1
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Quote:
Originally Posted by davar55 View Post
Construct a 5 x 5 square containing distinct primes
such that each row, column and diagonal sums to a distinct prime.
Is it a magic square where every sum is the same?
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Old 2008-04-30, 22:51   #3
davar55
 
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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.
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Old 2008-05-13, 18:07   #4
davar55
 
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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.)

Last fiddled with by davar55 on 2008-05-13 at 18:08
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Old 2008-05-17, 02:23   #5
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Here's one:
Code:
			      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

Last fiddled with by lavalamp on 2008-05-17 at 02:26
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Old 2008-05-17, 06:12   #6
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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
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Old 2008-05-17, 11:59   #7
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Hm, I think I worked the total out right, but wrote it down wrong.

It wasn't just a fluke, honest! ;)
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Old 2008-05-19, 14:01   #8
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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 ?
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Old 2008-05-19, 14:57   #9
lavalamp
 
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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:
			      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

Last fiddled with by lavalamp on 2008-05-19 at 15:06
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Old 2008-05-21, 12:54   #10
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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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