Last month, we transported puzzlers to the first day of autumn, when the probability that any given leaf would fall off a tree started at 1% per day and rose by 1% every day (so, on the first day it was 1%, on the second 2%, and so on) until all the leaves had fallen. If the tree started with 1,000,000 leaves, we asked, on what day will the greatest number of leaves fall off the tree, and what will that number of leaves be?

Solution

First of all, congratulations to this month’s winner, spreadsheet extraordinaire Orest Kuzemko who found that 62,816 leaves would fall on day 10.

The basic tradeoff at play was this:

• on day 1, there are many leaves on the tree but each one is very unlikely to fall off.
• by day 100, each remaining leaf is extremely likely to fall, but there aren’t many left.

So, there must be a balance point where the rate of leaf fall is greatest.

There were at least two ways to approach this problem. The first was to write a program (or spreadsheet) to simulate each day’s leaf fall, and do some accounting. The other was to pretend that leaves are continuous and use calculus to find the maximum.

The first was the easier, popular, and exact approach, so we’ll start there.

Accounting

The probability for a leaf to fall from the tree started at 1% on day 1, and rose by 1% each day. We can model the probability like $f_t=\frac{1}{100}t.$

If there are $L_t$ leaves left on day $t$, then the number of leaves that fall on that day will be $L_t\cdot f_t$.

This means that the number of leaves left on day $(t+1)$ is:

$$ \begin{align*}L_{t+1} &= L_t-L_t\cdot f_t \\ &= \left(1-f_t\right)L_t. \end{align*} $$

Which is a nice set of equation to set up in a spreadsheet: