Water Lily

Because of the water lily grows twice each day, so it must equal 19 because 20 minus 1 equals 19.

If today is the $20^{th}$ day then yesterday is the $19^{th}$ day. If today if full, then yesterday is half full since the size of the water lily is twice the previous day.

If the flower occupies the whole area of the pond in 20th day then on the 19th day it was half its size and hence it covered only half the area of the pond. It can be thus solved in a logical way rather than in mathematical way.

Let x be the size of the flower initially. Then, one can realize that the flower must double in size every day.

2x for the first day of growth.

2(2x) for the second day, and so on...

Finally, at day 20 we have:

$2^{20}x = A$

The question asks how many days it takes for half the pond's area to be occupied, so...

$\frac{A}{2} = 2^{19}x$

So the answer is 19 days.

Consider, the radius of the water lily is 1m and area of the circular pond is So in the 19th day the size of the water lily will be half area of the pond.