In mathematics, a P-multimagic square (also known as a satanic square) is a magic square that remains magic even if all its numbers are replaced by their kth powers for 1 ≤ k ≤ P. 2-multimagic squares are called bimagic, 3-multimagic squares are called trimagic, 4-multimagic squares tetramagic, and 5-multimagic squares pentamagic.

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.
Cubes 0, 1, 8, 27, 64, 125, 216, ... (sequence A000578 in the OEIS)
Sum of Cubes 0, 1, 9, 36, 100, ... (sequence A000537 in the OEIS)
Value for Trimagic squares 1, 50, 675, 4624, ... (sequence A052460 in the OEIS)

Similarly the tetramagic sequence
4-Power 0, 1, 16, 81, 256, 625, 1296, ... (sequence A000583 in the OEIS)
Sum of 4-Power 0, 1, 17, 98, 354, 979, 2275, ... (sequence A000538 in the OEIS)
Sums for Tetramagic squares 0, 1, 177, ... (sequence A052461 in the OEIS)

A bimagic square is a magic square that remains magic when all of its numbers are replaced by their squares.

The first known bimagic square has order 8 and magic constant 260 and a bimagic constant of 11180.

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.