Square of Primes

davar55 · 2008-04-30, 18:12

Construct a 5 x 5 square containing distinct primes
such that each row, column and diagonal sums to a distinct prime.

petrw1 · 2008-04-30, 22:35

Quote:

Originally Posted by davar55

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?

davar55 · 2008-04-30, 22:51

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.

davar55 · 2008-05-13, 18:07

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.)

lavalamp · 2008-05-17, 02:23

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

S485122 · 2008-05-17, 06:12

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

lavalamp · 2008-05-17, 11:59

Hm, I think I worked the total out right, but wrote it down wrong.

It wasn't just a fluke, honest! ;)

m_f_h · 2008-05-19, 14:01

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 ?

lavalamp · 2008-05-19, 14:57

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

davar55 · 2008-05-21, 12:54

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.

2008-04-30, 18:12	#1
davar55 May 2004 New York City 2³×23² Posts	Square of Primes Construct a 5 x 5 square containing distinct primes such that each row, column and diagonal sums to a distinct prime.

2008-05-13, 18:07	#4
davar55 May 2004 New York City 1088₁₆ Posts	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

2008-05-19, 14:57	#9
lavalamp Oct 2007 Manchester, UK 2²×5×67 Posts	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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2008-04-30, 22:51	#3
davar55 May 2004 New York City 2³×23² Posts	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.

2008-05-17, 06:12	#6
S485122 Sep 2006 Brussels, Belgium 13·127 Posts	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

2008-05-17, 11:59	#7
lavalamp Oct 2007 Manchester, UK 2²·5·67 Posts	Hm, I think I worked the total out right, but wrote it down wrong. It wasn't just a fluke, honest! ;)

2008-05-19, 14:01	#8
m_f_h Feb 2007 2⁴×3³ Posts	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 ?

2008-05-21, 12:54	#10
davar55 May 2004 New York City 2³·23² Posts	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.