Lily pads with a head start
A lily pad doubles its area every day. Starting with a single lily pad on day 1, the pond is exactly covered at the end of day 30.
If the pond instead starts with 8 such lily pads on day 1, at the end of which day is it first fully covered?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
How many doublings turn 1 pad into 8?
Starting with 8 pads is like starting with 1 pad a few days earlier.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Eight pads is pads: what one pad reaches after three doublings. So the head start saves exactly three days.
Step 1. One pad covers the pond after 30 doublings: the pond's area is times the pad's area.
Step 2. Starting with pads, after days the area is pads' worth. The pond is covered when , so .
So the pond is covered at the end of day 27.
The usual trap. Dividing by 8 gives about day 4. Exponential growth turns a multiplicative head start into an additive one: a factor saves days. In the same spirit, a pond covered on day 30 was half covered on day 29, not day 15.