Whats up /sci/ i came up with this problem when I was in my gf's bathroom, as the tiles in the wall have the pattern for the problem.

The problem (take a look at pic related to understand what Im talking about) is as follows:

Some preliminaries:
>A node is any connection between 2 or more lines
>A 2-node is specifically a connection between only 2 lines, same for the 3-node and the 4-node
>A square is, well, a square

For an odd number of starting squares:
1) find a formula for the total number of squares
2)find a formula for the total number of steps to the top
3) find a formula for the total number of nodes, 2-nodes, 3-nodes and 4-nodes
4)prove or disprove there will always be an even number of nodes
5)prove or disprove that the final row will always be composed of only 1 square
6)find a formula for the number of squares in the central column of squares (the column that starts with the top square)

For an even number of starting squares:
(1 thru 3 the same as for odd squares)
4) prove or disprove there will always be an odd number of nodes
5)prove or disprove that the final row will always have 2 squares
6)find a formula for the number of squares in each of the central columns

I dont know if this problem is brainlet tier and theres already formulas for it since im not a mathematitian but it seemed interesting.
Hope this turns into a nice, fun thread.