"Chaos" (The Movie) And The Pigeonhole Principle - An Interesting Goof

***Caution: Spoiler Ahead***

While reading the list of goofs in the movie Chaos, I noticed that nobody had pointed out this important mathematical goof in the movie. At around 1:13:00 time in the movie, note this set of dialogs

1. "...about a billion dollars..."
2. "Now. If you look at all the transactions, no 2 amounts are the same, and none of them are over 100 dollars"
3. "...less than 100 dollars, from, say, 10 million accounts, no reflex"

According to the Pigeonhole principle, it's not possible to make about 10 million transactions, each less than 100 dollars, such that no two transactions have the same amount [assuming the least count is 1 cent]. It's only possible if transaction limit is 100,000 dollars.

However, if 10 million transactions [all withdrawals] are made such that no two amount to the same, then the minimum amount withdrawn would be ((10,000,000)*(10,000,001)/2) cents - about 500,000,000,000 dollars- far, far more than 1 billion dollars.

Pigeonhole Principle - Also known as Drichlet's Box / Drawer Principle

It's unfortunate that the people behind Chaos overlooked this important mistake -especially when they were trying to make a movie that apparently is so dependent on mathematics and Chaos Theory.

Finally, to withdraw 1 billion dollars such that no 2 withdrawals amount to the same, we need at least about 450,000 withdrawals [starting from 1 cent, 2 cent, 3 cent and so on till about 450,000 cents].

A Goof In The Movie "Crimson Tide"