The Magic Encyclopedia ™

The Magic bigrade
{note: investigative article}
(by Aale de Winkel)

The magic bigrade (or 2-Multigrade) has some special properties this article intent to list

The bigrade
The magic bigrade is a line of m numbers out of the numbers [1 .. mn] with
k=0m-1Bk = m (mn + 1) / 2 and
k=0m-1Bk2 = m (2 m2n + 3 mn + 1) / 6
centralized sum changing numbers Bk --> (mn + 1) / 2 + bk it follows:
k=0m-1bk = 0 and
k=0m-1bk2 = m (m2n - 1) / 12
k=0m-1Bk = k=0m-1(mn + 1) / 2 + bk = m (mn + 1) / 2 + k=0m-1bk m (mn + 1) / 2 ==>
k=0m-1bk = 0

k=0m-1[(mn + 1) / 2 + bk]2 = k=0m-1[(mn + 1)2 / 4 + (mn + 1) * bk + bk2] =
m (mn + 1)2 / 4 + k=0m-1 bk2 = m (2 m2n + 3 mn + 1) / 6 ==>
k=0m-1 bk2 = m (2 m2n + 3 mn + 1) / 6 - m (mn + 1)2 / 4 = m (m2n - 1) / 12
bigrade move the move from one bigrade to another is given by B2k --> B1k + dk with
2 k=0m-1B1kdk = - 2 k=0m-1B2kdk = - k=0m-1dk2
k=0m-1B2k2 = k=0m-1(B1k + dk)2 = k=0m-1B1k2 + 2 k=0m-1B1kdk + k=0m-1dk2 ==>
2 k=0m-1B1kdk = - k=0m-1dk2 = - 2 k=0m-1B2kdk
The bigrade move make it possible to "hang" the square onto the first line with a set of differences
the twisted condition these difffernces is curiously bound to the first lines numbers, the validity
of these difference squares remain to be seen