Is this a mistake in The Imitation Game?

Question

In explaining how hard it is to crack the code of the Enigma machine, Cumberbatch' character says in The Imitation Game:

There are 159 million, million, million possible Enigma settings. All we had to do was try each one. But if we had 10 men checking one setting a minute for 24 hours every day and seven days every week, how many days do you think it would take to check each of the settings? Well, it’s not days; it’s years. It’s 20 million years. To stop an incoming attack, we would have to check 20 million years’ worth of settings in 20 minutes.

For no reason other than curiosity, I tried to calculate whether or not this "20 million years" is correct, and found that it is completely incorrect.

Let's just use simple math to do this: It's stated that the Enigma machine has 159 million million million different settings, that's 1.59×10^20. Let's assume that a man can check a code each second (which is much faster than what was stated in the movie), and while a year has 31,536,000 seconds, then in 20 million years period that would be: 10(men)× 31,536,000(seconds in a year) × 20 ×1,000,000 (20 million years) = 6.3×10^15 settings. Obviously that is so much less than the Enigma settings so there is a huge error in saying "20 million years" as it would take much much longer than that. 25,000 times more to be precise.

So is this a calculation mistake they made?

Answer 1

"There are 159 million, million, million possible Enigma settings":

• 159,000,000,000,000,000,000 settings

"10 men checking one setting a minute for 24 hours every day":

• 10 x 60 x 24 = 14,400 settings checked per day

"how many days do you think it would take to check each of the settings"?

• 159,000,000,000,000,000,000 / 14,400 = 11,041,700,000,000,000 days

"it’s not days; it’s years"

• 11,041,700,000,000,000 / 365 = 30,251,100,000,000 years or 30.25 trillion years

To the OP's point about checking one setting per second: that would result in 864,000 settings checked per day, and would take a total of about 500 billion years. So obviously the movie's "20 million years" was way, way too low.

Answer 2

I think your calculations are wrong, here is why; in your quote it says "All we had to do was try each one. But if we had 10 men checking one setting a minute"

with that said, it means that we put 10 men just to solve one setting in one single minute, so the number 10 should not be used in the calculations; we have 1.59×10^20 settings if we divide it by 60(minutes in one hour)(24 hour in one day)(365 day in one year) = 525600 minutes in one year, so to find out how much it would take to solve all of the 1.59×10^20 settings we do it simply by :

1.59×10^20/ 525600 = 3*10^14 years so to solve the enigma settings we need 300 million million years and not 20 million years.