拼图 |越来越多的睡莲

问：圆形水池中央有一朵美丽的睡莲。睡莲的大小每天增加一倍。整整 20 天后，百合将覆盖整个水池。睡莲多少天后会覆盖一半的水池？

答：因为睡莲每天都翻倍，20天后整个水池被覆盖，所以在前一天，19天后，水池将被覆盖一半。

Let the size of the lily be S in the beginning.

At day 2, the size of the lily = S x 2 = (2^1)(S) = 2(S)
At day 3, the size of the lily = 2(S) x 2 = (2^2)(S)= 4(S)
At day 4, the size of the lily = 4(S) x 2 = (2^3)(S) = 8(S)
At day 5, the size of the lily = 8(S) x 2 = (2^4)(S) = 16(S)
.
.
At day 19, the size of the lily = 131072(S) x 2 = (2^18)(S) = 262144(S)
At day 20, the size of the lily = 262144(S) x 2 = (2^19)(S) = 524288(S)

Therefore, the size of the pool = (2^19)(S)

Now, the size of half of the pool = (2^19)(S)/2 = (2^18)(S)
which is the size of the lily at the 19th day.

Hence, After 19 days, the water lily will cover half of the pool.

结论： 19天后，睡莲将覆盖一半的水池。