If the squares are normal, the constant for the power-squares can be determined as follows: Bimagic series totals for bimagic squares are also linked to the square-pyramidal number sequence is as follows :- Squares 0, 1, 4, 9, 16, 25, 36, 49, .... (sequence A000290 in the OEIS) Sum of Squares 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ... (sequence A000330 in the OEIS) )number of units in a square-based pyramid) The bimagic series is the 1st, 4th, 9th in this series (divided by 1, 2, 3, n) etc.
so values for the rows and columns in order-1, order-2, order-3 Bimagic squares would be 1, 15, 95, 374, 1105, 2701, 5775, 11180, ... (sequence A052459 in the OEIS) The trimagic series would be related in the same way to the hyper-pyramidal sequence of nested cubes.
It has been conjectured by Bensen and Jacoby that no nontrivial[clarification needed] bimagic squares of order less than 8 exist.
This was shown for magic squares containing the elements 1 to n2 by Boyer and Trump.
However, J. R. Hendricks was able to show in 1998 that no bimagic square of order 3 exists, save for the trivial bimagic square containing the same number nine times.
The proof is fairly simple: let the following be our bimagic square.
In 2006 Jaroslaw Wroblewski built a non-normal bimagic square of order 6.
Also in 2006 Lee Morgenstern built several non-normal bimagic squares of order 7.
Trimagic squares of orders 12, 32, 64, 81 and 128 have been discovered so far; the only known trimagic square of order 12, given below, was found in June 2002 by German mathematician Walter Trump.
A 4-magic square of order 512 was constructed in May 2001 by André Viricel and Christian Boyer.
[1] The first 5-magic square, of order 1024 arrived about one month later, in June 2001 again by Viricel and Boyer.
Another 5-magic square, of order 729, was constructed in June 2003 by Li Wen.