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

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

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

You can find more short problems, arranged by curriculum topic, in our short problems collection.