He found an excellent description of my life: "Everyone but you".Gambit37 wrote:Oh Jan, you tease!
Look at this:
1 2 3
2 1 4
3 4 1
It's a 3*3-matrix, but numbers are 1...4. Fill rows and columns with the missing numbers:
1 4 3 2
4 1 2 3
3 2 1 4
2 3 4 1
Now create two more matrices by *rotating the columns* (except the first one):
1 3 2 4 | 1 2 4 3
4 2 3 1 | 4 3 1 2
3 1 4 2 | 3 4 2 1
2 4 1 3 | 2 1 3 4
add to the columns 0, 4, 8, 12. The three matrices look now like this:
1 8 11 14 | 1 7 10 16 | 1 6 12 15
4 5 10 15 | 4 6 11 13 | 4 7 9 14
3 6 9 16 | 3 5 12 14 | 3 8 10 13
2 7 12 13 | 2 8 9 15 | 2 5 11 16
to those matrices, i add the standard matrix and the diagonal mirrored one:
1 2 3 4 | 1 5 9 13
5 6 7 8 | 2 6 10 14
9 10 11 12 | 3 7 11 15
13 14 15 16 | 4 8 12 16
what's the magic in it?
you will find that each number meets each other number in exactly one line in one of those 5 matrices.
we created the order four of this: http://en.wikipedia.org/wiki/Affine_pla ... eometry%29
The same way you create order 9 with that magic matrix.
1 2 3 4 5 6 7 8
2 4 9 1 8 5 3 6
3 9 8 7 4 2 5 1
4 1 7 2 6 8 9 5
5 8 4 6 9 3 2 7
6 5 2 8 3 7 1 9
7 3 5 9 2 1 6 4
8 6 1 5 7 9 4 3
There are still a lot of open questions around that theme.
I found the solution for order 8 and 9 around 1984 - they weren't found then.
I dropped them into my desk again and said to myself 'whatever they think - I *am* a mathematician.
And - I am sure - I am still the only one knowing that the problem can be reduced to such a magic matrix -
know how to build the matrix . and my program says that the next unknown - order 12 - does not exist.
When I die, this little bit of useless extra knowledge dies too.
So much for citations...
