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We know that, recently, personal computers make around $10^9$ calculations per second, and I'm just curious about how many calculations was able to compute the machine invented by Turing for breaking the enigma code.

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This is only a very rough, back of the envelope answer because there were several versions of the enigma and the Bombe, but according to the Wikipedia article on the British Bombe, one version of the Bombe could

run through all 17,576 possible positions for one rotor order was about 20 minutes

Since $17576 = 26^3$, it's referring to the three-rotor version of the enigma. That comes out to about 15 operations a second.


Rob is correct that computing speed progressed extremely rapidly at this point too, so my answer represents a basic answer about one single machine, and obviously there were many others of varying capability as time went on.

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    $\begingroup$ Things got faster very quickly: the Colossus built in 1944 to decrypt the output of the Lorenz machine is estimated to have equivalent performance to a PC with a 5.8MHz clock (see the link). $\endgroup$ – Rob Arthan Apr 14 '15 at 16:27
  • $\begingroup$ @RobArthan And by 1946, the ENIAC was faster than that (I think). $\endgroup$ – M T Apr 14 '15 at 16:28
  • $\begingroup$ @RobArthan I mentioned your comment in my answer because that's a great point. Thanks! $\endgroup$ – M T Apr 14 '15 at 16:31
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    $\begingroup$ Remember that the Bombe relied on actual mechanical rotors of its own to do its calculations, and the Colossus used a long loop of paper tape as an essential part of its mechanism. At least as late as 2013 (perhaps even today) the replica Colossus at Bletchley Park was still operational. The speed at which the tape runs over its reels is quite amazing. $\endgroup$ – David K Apr 14 '15 at 16:57

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