Lily Pad Doubling: Head Start Problem
A single lily pad floats in a pond. Each day, its area doubles until it covers the entire pond. Starting from one lily pad, the pond is fully covered on day 30.
If you instead start with 8 lily pads on day 1, how many days does it take to cover the entire pond?
Hints
- Think about when a single lily pad would naturally reach the count of 8 pads on its own doubling trajectory -- that tells you the head start.
- Since area doubles each day, $2^3 = 8$. Starting with 8 pads is equivalent to a 3-day head start over the single-pad scenario.
- The single-pad scenario reaches full coverage on day 30. A 3-day head start means the 8-pad scenario finishes on day $30 - 3 = 27$.
Worked Solution
How to Think About It: Doubling problems are all about logarithms and equivalence. The key question is: starting with 8 pads is the same as starting with 1 pad but at what day? Since the area doubles each day, 1 pad becomes 8 pads after 3 days ($2^3 = 8$). So starting with 8 pads today is equivalent to having started with 1 pad 3 days ago.
Quick Estimate: Day 30 covers the full pond with 1 starting pad. Three days of doubling get you from 1 to 8. So 8 pads should finish 3 days earlier: day $30 - 3 = 27$. That's the answer -- no algebra needed once you see the equivalence.
Formal Solution:
Let the pond's total area be $A$. Starting from 1 lily pad with area $a_0$, after $n$ days the total area is $a_0 \cdot 2^n$. The pond is covered when: $$a_0 \cdot 2^n = A \implies 2^n = A / a_0$$
We are told this happens at $n = 30$, so $A / a_0 = 2^{30}$.
Now start with 8 pads, each of area $a_0$. After $n$ days the total area is $8 a_0 \cdot 2^n = a_0 \cdot 2^{n+3}$. The pond is covered when: $$a_0 \cdot 2^{n+3} = A = a_0 \cdot 2^{30} \implies n + 3 = 30 \implies n = 27$$
Answer: 27 days.
Intuition
Exponential growth problems are solved by thinking in log space. Linear shifts in log space (adding 3 days) correspond to multiplicative shifts in linear space (multiplying by $2^3 = 8$). This is why doubling time is such a powerful heuristic -- once you know the doubling period, you can mentally shift any scenario forward or backward in time by counting doublings.
This same reasoning applies in finance: if an asset compounds at a fixed rate and you want to know how a larger starting capital changes the time to reach a target, you simply compute the log ratio of starting values divided by the log of the growth factor. The rule of 72 is another example -- it is just a quick approximation to this same log calculation.