In how many ways can a $4 \times 4$ square grid be coloured using four different colours so that no colour is repeated in any row, column, or along the two main diagonals. For clarity, one valid solution to this problem is shown below.
I am after unique solutions, so:
- rotations of the grid through angles of $90^\circ, 180^\circ$, and $270^\circ$, and
- reflections about the horizontal, vertical, and the two main diagonals
are not considered different.
I am guessing this problem is perhaps well known, so I apologise in advance, but have not been able to make much progress towards its solution. It may even go by a well known name, making it easier to identify, and if this is indeed the case I would be interested in knowing what it is called.