Bimagic Squares 9 x 9, Interactive Solution based on Ternary Matrices
Introduction
A Magic Square is bimagic if it remains magic after each of the numbers have been squared.
It has been proven that the smallest order of Bimagic Squares is 8.
The Magic Sum of the squared numbers is n(n^{2} + 1)(2n^{2} + 1)/6 = s1(2n^{2} +1)/3 with s1 the magic sum of the original square.
John Hendricks devellopped a Class of 9^{th} order bimagic squares  based on the ‘digital equation’ method  which can be decomposed into 4 regular grids based on the ternar representation of numbers (0 = 0000, 1 = 0001, 2 = 0002, 3 = 0010 ...
80 = 2222).
Any Magic Square of the 9^{th} order with the numbers 1 ... 81 can be written as
G1 +
3 * G2 +
9 * G3 +
27 * G4
+ [1],
where the matrices
G1,
G2,
G3 and
G4
 further referred to as grids  contain only the numbers 0, 1 and 2.
The Base Grids
G1,
G2,
G3 and
G4
 suitable for the construction of bimagic squares  can be viewed in the form below by selecting
G1,
G2,
G3 or
G4
with the left selection button and pressing the button 'Shw G'.
The grids can be put in random order and the numbers 0, 1 and 2 in each grid can be swapped.
The construction method based on the principles described above has been applied in following Interactive Solution:
Procedure:

Select the Ternary Matrices with the four upper left selection buttons and validate the selection with the button 'Validate'.

The selected Matrices can be viewed by selecting
G1,
G2,
G3 or
G4
with the left selection button and pressing the button 'Shw G'.

Define the sequence with the applicable selection buttons (select 1, 2 ... 4 for G1, G2 ... G4).

Press the button ‘Shw Sqr 1’ to calculate and visualise the resulting Bimagic Square.

Press the button ‘Shw Sqr 2’ to calculate and visualise the corresponding squared numbers.
The 4! = 24 permutations of 6^{4} = 1296 possible grids results in 24 * 1296 = 31104 possible Bimagic Squares.
Have Fun!
