A sequence of square mosaics is constructed with black and white square tiles, all of the same size, with the first being formed by a white tile surrounded by black tiles, the second by four white tiles surrounded by black tiles, and so on, as indicated in the figure. If in a sequence of mosaics formed according to this rule, 80 black tiles are used, how many white tiles will be used?
(a) 55
(d) 85
(b) 65
(e) 100
(c) 75

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-004.jpg?height=191&width=500&top_left_y=1646&top_left_x=1326)
The correct option is (a).

In the first mosaic, we have $3+3+1+1=8$ black tiles, in the second, we have $4+4+2+2=12$, in the third, we have $5+5+3+3=16$, and it is not difficult to see (and verify) that the next mosaics have 20 and 24 black tiles, as each new mosaic uses four more black tiles, one on each side. Since $8+12+16+20+24=80$, it is possible to construct exactly five mosaics. Finally, the total number of white tiles in this sequence of five mosaics is

$$
1^{2}+2^{2}+3^{2}+4^{2}+5^{2}=1+4+9+16+25=55
$$

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-039.jpg?height=226&width=892&top_left_y=815&top_left_x=614)