There's something irresistible about this kind of exercise, isn't there? On one hand, we all want a magical source of money, and on the other, the obvious answer isn't always the most convenient in the long run. With that in mind, we arrive at a classic question: One million dollars in cash vs. a penny that doubles every day. Which do you choose? While the effects of exponentiation are tempting, there's a huge underlying problem that not everyone considers…
Pennies and Grains of Rice
If the exercise sounds familiar, it's because you've come across it before, not in the form of millions of dollars and pennies, but rather with chessboards and grains of rice. The mysterious traveler arrives at the court of the king/emperor, presents the game of chess, and with the promise of a reward from the ruler, asks only for a grain of rice for the first square, two for the second, four for the third, and so on until reaching the last one.
In compact form, the traveler asked for 2^64 - 1 grains of rice, something like the world production of 1,200 years. Unfortunately we never knew if the traveler lost his head for his maneuver or if he received a place at court in exchange, but what brings us together here today is the modern variant of the legend, which involves one million dollars and a penny that doubles each day. What's our goal? In essence, to explain why you should choose the million dollars.
Million vs. Penny: The Weight Detail
The calculation is nothing strange:
- One dollar is 100 cents.
- One million dollars is 100 million cents.
- The first day we have one penny, the second two, the third four, and so on.
- We know that 2 raised to the 27 (2^27) gives us 134,217,728. If we think in days and cents, in less than a month we comfortably exceed one million dollars.
So exponential growth guarantees us more money. However…
- The official US penny weighs 2.5 grams.
- 100 million pennies x 2.5 grams = 250 million grams = 250 tons.
- 134,217,728 pennies x 2.5 grams = 335.5 million grams = 335.5 tons.
- And that number keeps doubling!
Therefore, in less than 90 days, the magic of that penny would turn you into hyper-millionaire, but the gigantic mass of zinc coated in copper would make the Earth collapse. Game over.
An Example... in Real Life?
To conclude, I want to share a recent story that, while it doesn't involve exponential growth, is very useful for visualizing the penny problem: Miles Walker is the owner of an auto repair shop in Georgia who currently has problems with several of his former employees and the US government itself. One of those former employees sued him for not receiving the last salary check, and Mr. Walker, who seems to have a very particular attitude, decided to make the payment by throwing a total of 91,500 pennies, soaked in oil, on the driveway of his garage.
The former employee in question and his partner had to spend almost seven hours collecting more than 228 kilograms of coins (remember, they were bathed in oil), and the total weight ended up bursting the wheels of their wheelbarrow. The Coinstar company collected and washed the coins, giving back in exchange a sum close to 915 dollars in paper. So, do you really want that magical penny...?
