The numbers 1 through 9 are arranged in a 3 x 3 grid so that each number is in the grid exactly once. They are arranged such that the top row plus the middle row gives the bottom row. If the grid forms another such addition when it is rotated 90 degrees to the left, what is its composition?

(Note: The numbers don't flip, for example 6 doesn't turn into 9.)

A                   B
A B C             C F I
D E F              B E H
G H I              A D G

Since A, D, G, H and I are all at some point formed by the addition of 2 numbers, and the smallest these numbers can be is 1 or 2, the numbers 1 and 2 must be out of B, C, E and F.

Neither A, D, C or B is 9, since there is no carrying from the hundreds column. Similarly, neither A nor D is 8 or 7, due to the fact that neither of them is 1 or 2.

So the range for both A and D is {3,4,5,6}. This means that G >= 7. Since G = A + D (+1 if carrying), A = C + B (+1) mod 10, and D = F + E (+1) mod 10; G must equal B + C + E + F (+3) mod 10.

This means that B + C + E + F must be greater than 14, since it cannot be less than 10 (minimum 1+2+3+4), and G must be 7 or greater. If 1 and 2 are in BCEF, the other 2 numbers must total 11 or more.

All I can concentrate on for now...

Posted by Iain on 2004-05-12 11:36:27