The Mathematics of the Movie "21"

## by Jeff Moehlis

### The movie "21" is the story of MIT students who "count cards" to improve their probability of winning the card game Blackjack at casinos. Not surprisingly, this movie has a lot of mathematics in it. Most obvious is the "counting of the cards", which is based on the techniques published in Edward O. Thorpe's 1962 book "Beat the Dealer". Discussions of the method and mathematics of "card counting" are described on various other websites. On this website, you can learn about other mathematical ideas which appear in the movie. I hope that this increases your enjoyment of the movie and perhaps teaches you some mathematics!

The Fibonacci Series

### In "21", when Ben Campbell (played by Jim Sturgess) is celebrating his birthday, the cake says 0, 1, 1, 2, 3, 5, 8, 13, ... These are the first terms in the Fibonacci Series, which was used as an example in the book Liber Abaci published in 1202 by Leonardo Fibonacci. This is obtained by first writing the numbers "0, 1", then defining each subsequent number as the sum of the previous two numbers in the series. Thus, the third number in the series is 1 = 1+0, the fourth number is 2 = 1+1, the fifth number is 3 = 2 + 1, etc. The next number on the cake would be 21=13+8, for Ben's 21st birthday. Clever, huh? (Hmmm, does "21" refer to Blackjack or Ben's age?) Ben will have to wait until he is 34 = 21+13 for his next "Fibonacci birthday". One can define other Fibonacci Series by specifying different numbers in the first two slots. For example, the Fibonacci Series starting with "2, 5" is 2, 5, 7, 12, 19, 31, 50, ...

The Monty Hall Problem

The Newton-Raphson Method

Counting Cards