It doesn't matter how you select your magic squares, because no matter which two magic squares you use, you will be able to produce a new magic square if you follow the 'multiplication' method. Draw a 4x4 square and go through the boxes one row at a time, left to right, top to bottom, counting from 1 to 16, but writing down the number of the box only when it falls on the diagonal. Let's see what happens if we try to use these quadruples for the columns of a 4x4 square.
Students can try many more such experiments.