I was doing some messing around on excel to answer this brain teaser:
In autumn the amount of leaves falling from a tree get doubled after every hour. The tree is leafless after 9 days. How many hours does it take until half of all leaves have fallen down?
WARNING: Brain teaser answer below, if you want to work it out yourself.
On Excel I was simulating doubling the number of leaves falling every hour, finding the cumulative leaves that have fallen, and then the proportion of the total leaves that have fallen (note 9 days = 216 hrs):
hour leaves falling cumlative proportion of leaves
that hour that have fallen
1 1 1 9.49557E-66
2 2 3 2.84867E-65
3 4 7 6.6469E-65
4 8 15 1.42434E-64
5 16 31 2.94363E-64
. . . .
211 1.6455E+63 3.29101E+63 0.03125
212 3.29101E+63 6.58202E+63 0.0625
213 6.58202E+63 1.3164E+64 0.125
214 1.3164E+64 2.63281E+64 0.25
215 2.63281E+64 5.26561E+64 0.5
216 5.26561E+64 1.05312E+65 1
It takes 215 hours for the first half to fall, and then the second half all fall in the final hour.
I can see this, but don't fully understand why this is.
What I can see is the second column is just the series $2^n$, where $n$ is the $hour-1$, and that the third column is one less than the next $2^n$, i.e.: $$ 2^{n+1} = 1 + \sum_{i=0}^{n}2^i $$
No doubt that, at hour 215, this 1 before the sum is so insignificant that: $$ 2^{n+1} \approx \sum_{i=0}^{n}2^i $$ for large $n$. I can see therefore that, for all intents and purposes, half of the leaves do fall in the last hour.
My question is, why does: $$ 2^{n+1} = 1 + \sum_{i=0}^{n}2^i $$ Is there any intuitive logic that can explain this? Or, if not, a mathematical proof?
