Three algorithms for construction of a classical center-symmetric magic matrix of
3;(6lk+m) size
Proposition 1. Let us introduce the matrices P = {pi, j}, A = {ai, j} and B = {bi, j}; 1 ; i ; 3;
1 ; j ; 6lk+m; k = 1, 2, ...; sign(x) = x/|x| if x ; 0 and sign(0) = 0; sg(x) = 1 if x > 0 and sg(x) = 0
otherwise; [x] — an integer part of x. And let
;) for l = 1 and m = 3
a1, j = – 3sign|1 – (j mod 3)| – sign|2 – (j mod 3)| – 2sign|3 – (j mod 3)|;
a2, j = – sign|1 – (j mod 3)| – 2sign|2 – (j mod 3)| – 3sign|3 – (j mod 3)|;
a3, j = – 2sign|1 – (j mod 3)| – 3sign|2 – (j mod 3)| – sign|3 – (j mod 3)|;
b1, j = (6k + 4 – j)·sign(j mod 2) + (6k – 1 – j)·sign|1 – (j mod 2)|;
b2, j = (6k – j)·sign(j mod 2) + (6k + 4 – j)·sign|1 – (j mod 2)|;
;) for l = 1 and m = 5
a1, j = 8 – sign(j – 1) – 2sign |j – 2| – 2sign |6k + 4 – j| – 3sign |6k + 5 – j| +
[(sg(j – 2) + sg(6k + 4 – j))/2]·(6 – 3sign(j mod 3) – 2sign|1 – (j mod 3)| –
sign|2 – (j mod 3)|);
a2, j = 8 – 3sign(j – 1) – 3sign |j – 2| – sign |6k + 4 – j| – sign |6k + 5 – j| +
[(sg(j – 2) + sg(6k + 4 – j))/2]·(6 – 2sign(j mod 3) – sign|1 – (j mod 3)| –
3sign|2 – (j mod 3)|);
a3, j = 8 – sign(j – 1) – 2sign |j – 2| – 3sign |6k + 4 – j| – 2sign |6k + 5 – j| +
[(sg(j – 2) + sg(6k + 4 – j))/2]·(6 – sign(j mod 3) – sign|1 – (j mod 3)| –
2sign|2 – (j mod 3)|);
b1, j = (6k + 6 – j)·sign(j mod 2) + (6k – j)·sign|1 – (j mod 2)|;
b2, j = (6k + 1 – j)·sign(j mod 2) + (6k + 6 – j)·sign|1 – (j mod 2)|;
;) for l = 2 and m = 7,
a1, j = 3 – 3sign |6k + 4 – j| + [(sg(j ) + sg(6k + 4 – j))/2]·(6 – sign|1 – (j mod 3)| –
2sign|2 – (j mod 3)| – 3sign(j mod 3)) + [(sg|6k + 4 – j| +
sg|12k + 8 – j|)/2]·(6 – 2sign|1 – (j mod 3)| – 3sign|2 – (j mod 3)| – sign(j mod 3));
a2, j = 6 – 2sign|1 – (j mod 3)| – 3sign|2 – (j mod 3)| – sign(j mod 3);
a3, j = 1 – sign |6k + 4 – j| + [(sg(j ) + sg(6k + 4 – j))/2]·(6 – 3sign|1 – (j mod 3)| –
sign|2 – (j mod 3)| – 2sign(j mod 3)) + [(sg|6k + 4 – j| +
sg|12k + 8 – j|)/2]·(6 – sign|1 – (j mod 3)| – 2sign|2 – (j mod 3)| – 3sign(j mod 3));
b1, 6k+4 = 6k – 2; b3, 6k+4 = 9k + 6;
b3 – (j mod 3), j = j;
for 1 ; p ; 3k + 1 and 1 ; q ; 3k + 2
b((p mod 3) + 1, 2p = 6k + 4 – p; b((p – 1) mod 3) + 1, 6k+4 + 2p = 6k – 2 – p;
b((1+q) mod 3) + 1, 2q – 1 = 12k + 8 – q; b(q mod 3) + 1, 6k+3 + 2q = 9k + 5 – q;
b((1+p) mod 3) + 1, 2p = 12k + 8 – p; b((p mod 3) + 1, 6k+4 + 2p = 9k + 6 – p;
b((q–1) mod 3) + 1, 2q – 1 = 6k + 5 – q; b((1+q) mod 3) + 1, 6k+3 + 2q = 6k – q.
Then any matrix P = {(6lk+m)·(ai, j – 1) + bi, j} is an example of the desired classical center-
symmetric rectangular matrix.
Ïîëó÷åíî ýëåêòðîííîé ïî÷òîé îò Þ.Â.×åáðàêîâà 9 èþíÿ 2015 ã.