Our lottery problem showed us just how much a common perception can diverge from the mathematical
reality of things! This next problem is going to take a similar exercise in perception, this time looking at how intuition alone isn’t always correct, or enough.
Formal, logical, mathematical proofs aren’t just something for theories or abstract mathematics, and the
more you study them, the more you find they’ll help you formalize your point of view—and help you improve your
perception of things, too! Just consider the following:
You’re at a party, perhaps offering a deal to a friend or acquaintance . . . or, better yet, a rich individual you’re not
too crazy about!
Here’s what you say:
“I’ll give you $25,000 one month from now, and all you have to give me in return is one penny today,
and double that penny every day for thirty additional days.”
Now, you’re going to want to have a contract and signed witnesses, because—well, we’ll get to that in just a bit.
Let’s say the individual in question agrees to your deal. He’ll give you one cent today, two cents tomorrow, four
the day after that, then eight cents—and so on. You’ve just given up $25,000! This is sounding like a pretty great deal for your friend, isn’t it?
He’ll end up with $25,000, and you’ll end up with—at most—a few hundred dollars.
What’s your perception of the deal? Is it a better deal for you, or your friend?
By now, you know enough to know that if it were a good deal for your friend, it wouldn’t even be mentioned in this book. Let’s take a look at the reality of how our little wager will turn out; consult the chart below. As with the challenge, we’ll take one penny and double it every day for thirty days:
Number of Days // Total Amount (Dollars)
0 $.01 = 1 cent
1 $.02 = 2 cents
2 $.04 = 4 cents
3 $.08 = 8 cents
4 $.16 = 16 cents
5 $.32 = 32 cents
6 $.64 = 64 cents
Take a look at how much the perception of our bet differed from the reality! After 15 days, the total is $327.68,
but after another extra 15 days? The total has increased to over $10 million! The first time I heard about this problem, I never imagined that doubling a penny for 30 days would turn into such a huge amount!
So, in this wager, you’re set to make over $10 million, and only have to give away $25,000. I don’t know about
you, but I’d take this deal any day!
What made the “perception” here differ so strongly? Well, that’s the power of exponentials! The term “exponential” is shorthand for many multiplications of the same number.
Imagine if you were to extend the deal to 100 days. How much money would you end up with? If you have an older
scientific calculator, try 2100 or 2 to the 100th power. This number is so large that the calculators give you back an
overflow error, because there are too many digits to display!
Actually, 2 to the 100th equals: $1,267,650,600,228,229,401,496,703,205,376—
which is a number with thirty-one digits. (You’re probably not going to be able to collect on this particular bet,
though—but it’s fun to imagine, isn’t it?)
The upshot here is that exponentials grow very quickly, as you saw in the chart, and many mathematical problems are of an exponential nature. Calculating these problems can become very difficult very quickly, since the numbers grow so fast that even computers are not fast enough to calculate the exponential number of combinations! But here’s an everyday example just for fun.
Say you have one cat and it costs you $25 per month to feed the kitty. That amounts to $300 per year. Not so bad.
Now let’s say another cat comes to live with you. Your food costs would now double to $600 per year. Okay, that seems doable, you reason. But let’s say that second cat was of the opposite gender of your first cat and now one of the cats has produced a litter of 6 kittens. And you just can’t give them up. They’re too cute! Now your original costs of cat food just jumped from $300 per year to $2,400 per year.
Now let’s say half of the newborn kittens are females, so each 3 of the female kittens now produce 6 new kittens
in a few months, so that’s another 18 new kittens to the current family of 8 kittens. There is now a total of 26 cats,
with the cost of 26 x $300 = $7,800 per year. I really hope I can use these cats as dependents on my taxes.
Keep this in mind: We’ll be coming back to exponents, and exponential problems several times in the book as we