To make a move, you would rotate a column or a row.
To rotate a column/row, you would shift the numbers over one position and wrap around a number from one end to the other while adding/subtracting 1 mod 6 to that wrapped around number only.
For example, maybe rotating a row left would add 1 mod 6 to that wrapped around number while rotating it right would subtract 1 mod 6.
The goal is to end up with the same number in all 5x5 locations.
More generally, you could have an nxn array with numbers in the range 0...k.
If k is low, you could use colors instead of numbers.
What do you think of this idea?