A water plant growing on a pond doubles the area it covers every day, and on the 30th day it covers the pond completely. On which day did it cover exactly half the pond?

Work backwards. Since the covered area doubles each day, the area on any day is half the area on the day after. The pond was full on day 30, so it was exactly half covered one day earlier, on day 29. The trap is day 15: halving the number of days is not the same as halving the area, because the growth is a doubling and not a steady addition — on day 15 the plant covers only about one thirty-thousandth of the pond.