Tiled Floor

Answer: 3025


The sequence of all floors

             

Even floors: 2 purple tiles per row (total number even)
Odd floors: 1 more purple tile than the previous floor (total number odd)
 

109 purple tiles
108 = 2$\times$54 so floor length 54 has 108 purple tiles
$\therefore$ floor length 55 has 108 + 1 = 109 purple tiles
Floor length 55 has 55$^2$ = 3025 tiles


The sequence of odd-sided floors
Each odd-sided floor contains the last odd-sided floor in the middle, outlined in blue.
4 purple tiles are added each time
Side length 2 longer each time

109 = 1 + 4$\times$?
      = 1 + 108
      = 1 + 4$\times$27

So side length = 1 + 2$\times$27 = 55

$\therefore$ total number of tiles = 55$^2$ = 3025


Finding the number of rows
There are 2 purple tiles on each row,
Except for the middle row which has 1 purple tile (on odd-sided floors)

109 tiles = 1 purple tile in the middle
                  + 108 purple tiles in pairs
              = 1 purple tile in the middle
                  + 54 pairs on 54 rows

$\therefore$ there are 1 + 54 = 55 rows
So there are 55$^2$ = 3025 tiles