A super number square has the following properties:

1. In each row, the rightmost number is the sum of the other three.
2. In each column, the bottom number is the sum of the other three.
3. Within each NW-SE 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).

(In reply to Solution by DJ)

As Dino has already observed, the square is symmetrical about the AFKQ diagonal. The question tells us that B=5, C=H, I=N. We can also deduce that Q=57 since the highest value in the grid is 57 and Q > all the other values (assuming that they are all positive integers). This transforms the grid to:

A 5 C D
5 F G C
C G K 5G
D C 5G 57

So we have the following equations:

A+5+C=D
5+F+G=C
C+G+K=5G
D+C+5G=57
A*F*K=57

The only factors of 57 are 1,3 and 19, therefore {A,F,K} = {1,3,19}. Which also gives:

A+F+K=23

Now we have 6 equations and 6 unknowns. By numerous substitutions through the above equations it's possible to get to:

2A-3F=35
3G=28-A ie G=9-(A-1)/3

The first of these shows that A>F, therefore A = 3 or 19.
The second equation shows that (A-1) must be divisible by 3.
Therefore the two equations show that A=19.

Substituting this back through the equations gives the same grid that Charlie obtained with his program.

Posted by fwaff on 2003-08-20 08:07:52