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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?

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    Your own quote says "10 men checking one per minute"... Of course if you do one per second it's going to be vastly different. – Catija May 13 '15 at 7:21
  • What @Catija said. – bobbyalex May 13 '15 at 8:00
  • So we have 159 million million million minutes / 60 / 24 / 365 = 302.5 million million years. Cheers maths – user25948 Sep 15 '15 at 14:35
  • The Turing machine was ingenious partly because it eliminated millions of terms that would not be used ever right off the bat. The license is used in Imitation Game for sense of inability for a non-computing machine to ever make a difference... – meh Feb 2 '17 at 21:18
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"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.

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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.

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    I think the problem described in the movie meant to indicate that each of the 10 men was checking a different setting -- so the group of 10 men as a whole would check 10 settings per minute, not just 1 setting per minute. (Because as you mention, if all 10 of the men are checking just one setting, there would be no need to mention that there were 10 of them.) – Shiz Z. May 13 '15 at 21:39
  • No i disagree with you he mentioned the number 10 men checking one settings to show us how dificult is to crack an enigma setting – Mourad May 14 '15 at 7:23
  • That makes sense, Mourad -- you are probably right -- I haven't seen the movie, but maybe it shows that it really does take 10 men to check one setting. In either case, I think we agree the movie's "20 million years" is way too low. – Shiz Z. May 14 '15 at 13:10
  • Yes Shiz its way way too low – Mourad May 14 '15 at 13:33

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