A square floor is tiled, as partially shown, with a large number of regular hexagonal tiles. The tiles are coloured blue or white. Each blue tile is surrounded by 6 white tiles and each white tile is surrounded by 3 white and 3 blue tiles. Ignoring part tiles, the ratio of the number of blue tiles to the number of white tiles is closest to
(A) $1: 6$
(D) $1: 4$
(B) $2: 3$
(E) $1: 2$
(C) $3: 10$

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-098.jpg?height=342&width=463&top_left_y=1252&top_left_x=1319)
A square floor is tiled, as partially shown, with a large number of regular hexagonal tiles. The tiles are coloured blue or white. Each blue tile is surrounded by 6 white tiles and each white tile is surrounded by 3 white and 3 blue tiles. Ignoring part tiles, the ratio of the number of blue tiles to the number of white tiles is closest to
(A) $1: 6$
(B) $2: 3$
(D) $1: 4$
(E) $1: 2$

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-425.jpg?height=336&width=442&top_left_y=325&top_left_x=1297)

## Solution

Let's start by considering seven tile configurations made up of one blue tile surrounded by six white tiles. If we look just at this tiling only in this way, it appears that there are six times as many white tiles as blue tiles. However, each white tile is adjacent to three different blue tiles. This means that every white tile is part of three different seven tile configurations. Thus, if we count white tiles as simply six times the number counted we will miss the fact that each white tile has been triple counted. Hence the number of white tiles is six times the number of blue tiles divided by three, or twice the number of blue tiles. The ratio of the number of blue tiles to the number of white tiles is 1:2.

ANSWER: (E)