has successfully built up her custom ceramic floor tile company, Tri-gon,
into a major corporation. She started Tri-gon by making and selling a unique
triangular tile 10cm on a side. At her new headquarters campus, the halls,
meeting rooms, cafeteria, laboratories, and plazas will have large,
triangular tile areas on the floor made with the original tiles.
Furthermore, each triangular area will have a one tile wide solid border
with the inside a different color. Because these areas are different sizes,
and she is planning further expansion in the future, she has
hired you as the architect to develop the formula and equation that will
allow her to know how many tiles are required for the border, and how many
are required for the interior. She did not get to be CEO of a large
corporation that she built from scratch by taking things on faith. You will
need to convince her that your solution will work for any size of tile area.
Her office has the large pattern below, with 39 tiles in the border and 25
in the interior, and her desk has the small pattern embedded, with 15 tiles
in the border and 1 tile in the interior.