Have I mentioned that I really like my math prof? He opened today’s class as follows (don’t forget the Russian accent in your mental soundtrack, and the grammar is his, not mine.):

“Okay! This story, I heard as a schoolboy. There was a man who was very lazy. He wanted money, but he did not want to work. One day, he came up to a bridge across a river, and the Devil appeared. The Devil offered to double his money every time he crossed the bridge, if the man would pay him $24 for each trip. So! The man was stupid as well as lazy, so he says yes!”

“The man cross the bridge once, and his money doubles! So he pays the Devil $24, and cross again. His money doubles again, so he pays the Devil $24. He cross third time, and the Devil takes the last $24 and disappear. How much did he have at beginning if the Devil took his last $24 on the third time?”

First, set a_{n} = amount of $ the guy has after he crosses the bridge.

We know that a_{3} = 0, and we want to find a_{0}, where a_{0} = his starting cash.

Becase we know his money doubles AND he loses $24 after crossing the bridge, we can say that:

a_{n+1} = 2a_{n} – 24

And then we work backwards to get:

a_{n} = (a_{n+1} + 24)/2 = (a_{n+1}/2) + 12

Which gives us:

a_{3} = 0

a_{2} = 12

a_{1} = 18

a_{0} = 21

This means our guy started out with $21 in his pocket.

So the moral of the story is that you need to study combinatorics so that you will know that the Devil is a cheat and so that you will be able to confuse him thoroughly when you tell him so. Or something. 😉