<image>In the process of constructing a certain venue, to beautify the floor, square tiles of the same specification in black and white colors are used to pave the rectangular floor. Please observe the following images and answer the related questions.
 
$$(1)$$ In the $$n$$-th diagram, each row has ______ tiles, the first column has ______ tiles, and the $$n$$-th diagram has ______ tiles in total $$(expressed in terms of an algebraic expression containing $$n$$)$$.
$$(2)$$ According to the above paving scheme, a rectangular floor required $$506$$ tiles in total. Find the value of $$n$$ in this case.
$$(3)$$ Is there a situation where the number of black tiles is equal to the number of white tiles? If it exists, find the value of $$n$$; if it does not exist, explain the reason through calculations.