A super number square has the following properties:
 In each row, the rightmost number is the sum of the other three.
 In each column, the bottom number is the sum of the other three.
 Within each NWSE diagonal line, the last number (bottom rightmost) is the product of the other numbers.
For example, if you have a square that looks like:
A B C D
E F G H
I J K L
M N P Q
you know that A+B+C=D, C+G+K=P, AFK=Q, EJ=P, and so on.
Construct a super number square in which the highest number in any position is 57, and the second number in the top row is a 5 (all numbers are positive integers).
Assuming that C must in fact be equal to H and I equal to N, we know
p=ej
l=5g
i=5gkj
n=5gkj
h=ejkg
c=ejkg
n=5gkj
f=ejk2ge
f=5gk2j5
from these last two, 7g=ej+2je+5
Thus various combinations of e and j produce what may or may not be valid (integral) g. We can then try various combinations of a and k to see if the product q is equal to d+h+l and to m+n+p where d=a+b+c and m is a+e+i.
Not wanting to do this by hand, I made a program:
CLS
sum = 2
DO
FOR j = 0 TO sum
e = sum  j
g = (e * j + 2 * j  e + 5) / 7
IF g = INT(g) AND g >= 0 THEN
FOR k = 0 TO 50
FOR a = 0 TO 50
IF e * j  k  2 * g  e > 0 AND 5 * g  k  j > 0 THEN
IF a * (e * j  k  2 * g  e) * k = a + 5 + 2 * (e * j  k  g) + 5 * g THEN
IF a + e + 5 * g  k  j + 5 * g  k  j + e * j = a * (e * j  k  2 * g  e) * k THEN
PRINT USING "####"; a;
PRINT USING "####"; 5;
PRINT USING "####"; e * j  k  g;
PRINT USING "####"; a + 5 + e * j  k  g
PRINT USING "####"; e;
PRINT USING "####"; e * j  k  2 * g  e;
PRINT USING "####"; g;
PRINT USING "####"; e * j  k  g
PRINT USING "####"; 5 * g  k  j;
PRINT USING "####"; j;
PRINT USING "####"; k;
PRINT USING "####"; 5 * g
PRINT USING "####"; a + e + 5 * g  k  j;
PRINT USING "####"; 5 * g  k  j;
PRINT USING "####"; e * j;
PRINT USING "####"; a * (e * j  k  2 * g  e) * k
PRINT ""
DO: LOOP UNTIL INKEY$ > ""
END IF
END IF
END IF
NEXT
NEXT
END IF
NEXT
sum = sum + 1
LOOP
the first several resulting arrays are:
21 5 11 37
5 3 3 11
11 3 1 15
37 11 15 63

19 5 9 33
5 1 3 9
9 3 3 15
33 9 15 57

11 5 15 31
5 6 4 15
15 4 1 20
31 15 20 66

9 5 10 24
5 1 4 10
10 4 6 20
24 10 20 54

of which the second has the desired highest number of 57.

Posted by Charlie
on 20030819 15:53:55 