The question you opened
The pond that fills on day 18
Algae covers a pond, doubling every day. On day 18 the pond is fully covered.
On which day is the pond half covered?
Day 17
One step back from full is one halving. Day 18 is full, so day 17 is half and day 16 is a quarter.
Algae doubles every day and covers the whole pond on day 18. The half covered day is day 17, and the popular first answer of day 9 is off by a factor of about 500. The logic is short. Doubling forward means halving backward. If the pond is full on day 18, then one day earlier it held half of that, because the next day doubled it to full. So day 17 is half. Day 16 is a quarter.
Day 15 is an eighth. Day 9 feels right because nine is half of eighteen, and we are trained to read halfway through the time as halfway through the process. That works for anything growing at a steady rate, a tap filling a bath, a car covering miles. It fails completely for anything that multiplies. Run day 9 forward and see how far off it is. From day 9 to day 18 is nine doublings, which multiplies by 512.
So the coverage on day 9 was one 512th of the pond, roughly two tenths of one percent. You would stand at the edge on day 9, see a patch of green the size of a dinner plate, and conclude the pond had years left. That last sentence is the whole point of the question. Exponential growth spends most of its life looking like nothing is happening, then finishes in a rush. Half the job gets done in the final step.
Three quarters of it gets done in the final two. The habit worth building is to ask what the growth multiplies by, then count backwards from the end rather than forwards from the start. Backwards from full is one halving per day, and the arithmetic becomes trivial. The same shape governs compound interest, viral spread, storage filling up and a bank balance that doubles every seven years. In each of them, the moment it looks urgent is already very near the end.
Technique: Run the doubling backwards