 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Prime magic (Posted on 2018-09-29) What is the smallest possible constant of a 3×3 magic square containing nine distinct non-composite numbers only?

 No Solution Yet Submitted by Ady TZIDON No Rating Comments: ( Back to comment list | You must be logged in to post comments.) possible solution | Comment 1 of 4

I think this problem relies a bit on the definition of a magic square and the word "small".

If a magic square elements are limited to the numbers 1, 2, 3, ..., n^2, then it is termed a "Normal" Magic Square.

The only solution (constant = 15) to the 3x3 for a Normal Magic Square are

8 1 6

3 5 7

4 9 2

(and its 7 rotations/reflections)

Additional squares may be generated by adding or multiplying the same constant to each term.

Composites, on the other hand, are all positive non-primes excluding 1. Clearly the Normal 3x3 is not made exclusively of non-composites.

Then we go for the entire set of 3x3s including the non-normal 3x3s.

All 3x3s with distinct integers are of the following form, as per Edouard Lucas:

For integers a,b,c, with 3c being the constant, these are of the form:

(-b+c) (a+b+c) (-a+c)

(-a+b+c) (c) (a-b+c)

(a+c) (-a-b+c) (b+c)

The normal solution has (a, b, c) = (-1, -3, 5), with constant = 15

So. with the broader definition, the terms now can become negative and the constant as well.

If  a constant being small means absolute value small, then a good candidate is

constant =  -3

17    -39     19

1     -1     -3

-21     37    -19

or the negative of this.

Here is a program and some results. The constant (row sum) is given first.

-60

-1    -59      0

-19    -20    -21

-40     19    -39

-57

-1    -57      1

-17    -19    -21

-39     19    -37

-45

3    -53      5

-13    -15    -17

-35     23    -33

-21

11    -45     13

-5     -7     -9

-27     31    -25

-3

17    -39     19

1     -1     -3

-21     37    -19

-60

-3    -57      0

-17    -20    -23

-40     17    -37

-54

-1    -55      2

-15    -18    -21

-38     19    -35

etc...

program ms

implicit none

integer a,b,c,j,con,ielt,flag,sign(3,9),term(9)

data sign/0,-1,1,   1,1,1,   -1,0,1,

1         -1,1,1,   0,0,1,   1,-1,1,

2         1,0,1,   -1,-1,1,   0,1,1/

do  a=-20,20

do b=-20,20

do 1 c=-20,20

do ielt=1,9

term(ielt)=sign(1,ielt)*a+sign(2,ielt)*b+sign(3,ielt)*c

call not_comp(term(ielt),flag)                        if(flag.eq.0)go to 1

do  j=1,ielt-1

if(term(ielt).eq.term(j))go to 1

enddo

enddo

con=3*c

print 3,con

3       format(/,i5)

print 2,term

2                  format( 3(i4,3x))

1                  enddo

enddo

enddo

end

subroutine not_comp(i,flag)

c Not composite means a positive integer not prime or 1

c Flag = 1 if true

implicit none

integer i,j,k,l,m,flag

flag=1

if (i.le.1)return

k=sqrt(1.*i)

do j=2,k

m=(1.*i)/(1.*j)

l=m*j

if(l.eq.i)go to 1

enddo

flag=1

return

1       flag=0

return

end

Edited on September 29, 2018, 7:41 pm
 Posted by Steven Lord on 2018-09-29 11:55:54 Please log in:

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